mathematical physics, analysis and geometry - volume 8

Mathematical Physics, Analysis and Geometry (2005) 8: 1–39 © Springer 2005DOI: 10.1007/s11040-004-1670-2

Symplectic Structures for the Cubic SchrödingerEquation in the Periodic and Scattering Case �

K. L. VANINSKYDepartment of Mathematics, Michigan State University, MI 48824, East Lansing, U.S.A.e-mail: [email protected]

(Received: 20 October 2001; in final form: 30 September 2003)

Abstract. We develop a unified approach for construction of symplectic forms for 1D integrableequations with the periodic and rapidly decaying initial data. As an example we consider the cubicnonlinear Schrödinger equation.

Mathematics Subject Classifications (2000): 35Q53, 58B99.

Key words: nonlinear Schrödinger, symplectic.

1. Introduction

1.1. GENERAL REMARKS

The main technical tool for the study of soliton systems is commutator formalism.All fashionable soliton systems like the Korteveg–de Vriez equation (KdV), thecubic nonlinear Schrödinger equation (NLS), the sin-Gordon equation, the Todalattice, etc., have such representation. Within the commutator formalism approachthe dynamical system appears as a compatibility condition for an over-determinedsystem of equations. As an example, we consider the NLS equation with repulsivenonlinearity��

iψ• = −ψ ′′ + 2|ψ |2ψ,

where ψ(x, t) is a complex function of spatial variable x and time t . The flow is acompatibility condition for the commutator

[∂t − V3, ∂x − V2] = 0,

with

V2(x, t) = V = − iλ

2σ3 + Y0 = − iλ

2

(1 00 −1

)+

(0 ψ

ψ 0

)

� The work is partially supported by NSF grant DMS-9971834.�� Prime ′ signifies derivative in variable x and dot • in time.

2 K. L. VANINSKY

and

V3(x, t) = λ

2

2

iσ3 − λY0 + |ψ |2iσ3 − iσ3Y′0.

The corresponding auxiliary linear problem

(∂x − V )f = 0, f =(

f1

f2

)

can be written in the form of an eigenvalue problem for the Dirac operator

Df =[(

1 00 −1

)i∂x +

(0 −iψ

iψ 0

)]f = λ

2f .

Another important feature of soliton systems is the Hamiltonian formulation.Here we assume that the potential ψ(x, t) is 2l-periodic: ψ(x + 2l, t) = ψ(x, t).For instance, the NLS flow can be written as

ψ• = {ψ, H3},with Hamiltonian H3 = 1

2

∫ l

−l|ψ ′|2 + |ψ |4 dx = energy and bracket

{A, B} = 2i

∫ l

−l

∂A

∂ψ(x)

∂B

∂ψ(x)− ∂A

∂ψ(x)

∂B

∂ψ(x)dx.

The bracket is nondegenerate. The corresponding symplectic form (up to a scalar)is:

ω0 = 2i〈δψ ∧ δψ〉, 〈•〉 = 1

2l

∫ l

−l

dx.

A priori it is not clear why the dynamical system, which arises as a compatibilitycondition has a Hamiltonian formulation. To put it differently, is it possible toobtain Hamiltonian formalism from the spectral problem?

Here we would like to make some historical remarks. Originally, the Hamil-tonian formulation of basic integrable models was found as an experimental fact.For the KdV equation the computation of symplectic structure in terms of thescattering data was performed by Faddeev and Zakharov [5]. It involved somenontrivial identities for the products of solutions. Later Kulish and Reiman [14]noted that all higher symplectic structures also can be written in terms of the scat-tering data. Again, they used the scheme of [5] and explicit calculations. Finally,we note that Zakharov and Manakov [28] for the NLS equation adopted a differentapproach. Instead of the symplectic structure they worked with the correspondingPoisson bracket. Again, using explicit formulas for the product of solutions theycomputed the Poisson bracket between the coefficients of the scattering matrix. Anappearance of explicit formulas that are the moving force of all these computations

SYMPLECTIC STRUCTURES FOR THE CUBIC SCHRÖDINGER EQUATION 3

seems to be quite mysterious. This was already discussed in the literature [4], anddescribed as a “computational miracle”.

The standard assumption needed to carry out spectral analysis is that the po-tential either is periodic or has rapid decay at infinity. We refer to the latter caseas scattering. Recently, in connection with the Seiberg–Witten theory [23, 24],Krichever and Phong [13] developed a new approach for the construction of sym-plectic formalism. The latest exposition of their results can be found in [3]. Themain idea of the Krichever–Phong approach is to introduce in a universal way thetwo-form on the space of auxiliary linear operators. This form is written in termsof the operator itself and its eigenfunctions. The goal of this paper is to review theKrichever–Phong approach in the case of 1D periodic NLS and to extend it to thescattering case. Within the unified approach, we reduce the number of formulasand eliminate unnecessary explicit computations. For instance, computation of thesymplectic form in terms of the spectral data (both in the periodic and the scatteringcase) becomes an application of the Cauchy residue theorem.

1.2. THE PERIODIC CASE

We assume that the potential is periodic with the period 2l: ψ(x +2l, t) = ψ(x, t).The Krichever–Phong formula, in the NLS context, takes the form

ω0 =∑P±

res〈e∗JδV ∧ δe〉 dλ.

This formula defines a closed 2-form ω0 on the space of operators ∂x − V (x, λ)

with 2l periodic potential. The set-up for this formula is broadly as follows.�

The eigenvalue problem

[J∂x − JV (x, λ)]e(x, λ) = 0, J = iσ2 =(

0 1−1 0

)

has special solutions, so-called Floquet solutions determined by the property e(x +2l, λ) = w(λ)e(x, λ). The complex constant w(λ) is called a Floquet multiplier.For each value of the spectral parameter λ there are two linearly independentFloquet solutions and two distinct Floquet multipliers. These solutions and cor-respondingly multipliers become single-valued functions of a point on the two-sheeted covering of the plane of spectral parameter λ. The simple points of the pe-riodic/antiperiodic spectrum of the eigenvalue problem constitute branching pointsof the cover. We assume that there is a finite number of simple points (so-calledfinite gap potential).

This two sheeted covering constitutes a hyperelliptic Riemann surface � withtwo infinities P+ and P− (Figure 1). Each point Q = (λ, ±) of � is specified bythe value of spectral parameter λ and the sheet “+” or “−” which determines the

� We refer to Section 2 for detailed discussion.

4 K. L. VANINSKY

Figure 1. Smooth Riemann surface �.

Floquet multiplier w(Q) corresponding to this λ. At every point of the curve wealso have a Floquet solution e(x, Q) which becomes a function of the point Q andsatisfies the identity e(x + 2l, Q) = w(Q)e(x, Q). The Floquet solution e(x, Q)

has an exponential singularity at infinities and plays the role of so-called Baker–Akhiezer function for the curve �.

At every point of the curve � we can define another solution e∗(x, Q). This isthe Floquet solution which is brought from a point on the different sheet but withthe same value of the spectral parameter λ. It is transposed and suitably normalized.The operator J∂x − JV (x, λ) acts on the solution e∗(x, Q) as an adjoint, i.e. onthe right:

e∗(x, Q)[J∂x − JV ] = 0.

It is assumed that the phase space consists of smooth 2l-periodic functions ψ(x)

or equivalently operators ∂x − V (x, λ) with 2l-periodic potential. The NLS flowacts on this space as it acts on the space of functions ψ(x). All notions of differen-tial geometry with obvious conventions can be applied to this space of operators.On the space of potentials we have a variation δψ(x). Thus for a fixed value ofthe spectral parameter λ we have well defined variation δV (x, λ). The variationsδe(x, Q), δe∗(x, Q) are defined correctly when λ = λ(Q) is fixed. Therefore,at each point Q of the surface � we have well defined meromorphic in Q thetwo-form

〈e∗JδV ∧ δe〉 dλ.

It takes values in the space of skew-symmetric two-forms on the space of operators∂x − V . The result of Krichever and Phong states that the sum of residues of thisform at infinities P± is nothing but the symplectic form ω0.

The formula has a lot of good properties. First, it produces all higher symplecticstructures by introducing the weight λn under the residue

ωn =∑P±

res λn〈e∗JδV ∧ δe〉 dλ, n = 1, 2, . . . .


Second, it easily leads to the Darboux coordinates, or in physics terminology theseparation variables, see Sklyanin [25]. These are local coordinates where thesymplectic form ω0 takes the simple canonical form

ω0 = 2

i

∑k

δp(γk) ∧ δλ(γk).

This merits special explanation. It is well known since the work of Flashka andMcLaughlin [6], that the poles γk of Floquet solutions lead to the Darboux co-ordinates for symplectic forms.� Recently, a lot of work was performed [15] toconstruct such variables for the Ruijsenaars–Sneider and the Moser–Calogero sys-tems. This required formidable technical machinery and extensive computations.At the same time, as it was demonstrated by Krichever [11], the formula leads tothe same result only by applying the Cauchy residue theorem.

1.3. THE SCATTERING CASE

The main goal of the present paper is to show that suitably interpreted the newapproach can be adopted for soliton systems with rapidly decaying initial data onthe entire line. This is the so-called scattering case.��

For such potentials one can define so-called Jost solutions J±(x, λ). These arematrix solutions of the auxiliary linear problem J ′± = V J± with the asymptoticsJ ′±(x, λ) = exp (−i λ

2xσ3) + o(1), as x → ±∞. Their columns J± = [j (1)± , j

(2)± ]

are analytic in the corresponding upper/lower half-plane.Our construction of the associated Riemann surface �∞ is a geometrical inter-

pretation of what is called the Riemann–Hilbert approach to the scattering problem,see [4]. A singular curve �∞ is obtained by taking two copies of the complex planeand gluing them to each other along the real line (Figure 2). The curve �∞ has twoinfinities P+ and P− and continuum set of singular points above the real line. Thestandard Jost solutions are lifted on �∞ and become the single valued function of apoint on the curve. Different branches of BA function are connected along the realline by the scattering matrix S:

S(λ) = 1

a

[1 b

−b 1

].

The Jost solution has exponential singularity at infinities and plays the role of theBaker–Akhiezer function for the curve �∞. This construction is explained in detailin Section 3.

The formula of Theorem 3.4 looks similar to the periodic case

ω0 = trace res 12 [〈H ∗+JδV ∧ δH+〉 + 〈H ∗−JδV ∧ δH−〉] dλ.

� See also Novikov and Veselov [22], for general discussion.�� We refer to Section 3 for detailed definitions.

6 K. L. VANINSKY

Figure 2. Singular Riemann surface �∞.

The only difference now is that we work with the matrix solutions

H+(λ) = [j

(1)− (λ), j

(2)+ (λ)

]and H−(λ) = [

j(1)+ (λ), j

(2)− (λ)

]and H+

+ (λ) = σ1HT+ , H+

− (λ) = σ1HT− , with

σ1 =(

0 11 0

).

The averaging now corresponds to the integration on the entire line

〈•〉 =∫ +∞

−∞dx.

The residue can be computed explicitly ω0 = 2i〈δψ ∧ δψ〉. Theorem 3.6 statesthat the symplectic structure can be put in the Darboux form

ω0 = 1

πi

∫ +∞

−∞δb(λ) ∧ δb(λ)

|a(λ)|2 dλ,

where a and b are coefficients of the scattering matrix S. Again identically tothe periodic case this result is obtained by applying the Cauchy residue theorem.Only now the sum of the residues in the affine part of the curve is replaced by itscontinuous analog. This is the integral which stays in the right hand side of theformula.

The unified approach to construction of symplectic forms produces an interest-ing problem. As we see, the symplectic form constructed in the periodic case hastwo systems of Darboux coordinates. One system is associated with poles of theFloquet solution. It is the divisor-quasimomentum Darboux coordinates. Anothersystem of Darboux coordinates is the action-angle variables. At the same time inthe scattering case we know only one system of Darboux coordinates. These areaction-angle variables. What is the correct analog of the divisor-quasimomentumin the scattering case? This is a subject of future publication [27].


2. The Periodic Case

2.1. THE DIRECT SPECTRAL PROBLEM

We provide here information needed in the next section for construction of sym-plectic forms. We refer to classical books [20, 21] for standard facts of spectraltheory and algebraic-geometrical approach to solitons.

The NLS equation

iψ• = −ψ ′′ + 2|ψ |2ψ, (2.1)

where ψ(x, t) is a smooth complex function 2l-periodic in x, is a Hamiltoniansystem

ψ• = {ψ, H},with the Hamiltonian H = 1

2

∫ l

−l|ψ ′|2 + |ψ |4 dx = energy and the bracket

{A, B} = 2i

∫ l

−l

∂A

∂ψ(x)

∂B

∂ψ(x)− ∂A

∂ψ(x)

∂B

∂ψ(x)dx.

The NLS Hamiltonian H = H3 is one in the infinite series of conserved integralsof motion.

H1 = 1

2

∫ l

−l

|ψ |2 dx,

H2 = 1

2i

∫ l

−l

ψψ ′ dx,

H3 = 1

2

∫ l

−l

|ψ ′|2 + |ψ |4 dx, etc.

These Hamiltonians produce an infinite hierarchy of flows etXm , m = 1, 2, . . . .The first in the hierarchy is the phase flow etX1 generated by the vector field

X1: ψ• = {ψ, H1} = −iψ.

The phase flow is a compatibility condition for

[∂t − V1, ∂x − V2] = 0, (2.2)

with� V1 = i2σ3 and

V2 = − iλ

2σ3 + Y0 =

(− iλ2 0

0 iλ2

)+

(0 ψ

ψ 0

).

� Here and below σ denotes the Pauli matrices

σ1 =(

0 11 0

), σ2 =

(0 −i

i 0

), σ3 =

(1 00 −1

).

8 K. L. VANINSKY

We often omit the subscript V = V2. The second, translation flow etX2 generatedby

X2: ψ• = {ψ, H2} = ψ ′

is equivalent to (2.2) with V1 replaced by V2. Finally, the third, original NLSflow (2.1) is a compatibility condition for (2.2) with V1 replaced by

V3 = λ2

2iσ3 − λY0 + |ψ |2iσ3 − iσ3Y

′0.

All flows of infinite hierarhy etXm , m = 1, 2, . . . commute with each other

[∂τm− Vm, ∂τn

− Vn] = 0.

The first times τ1, τ2 and τ3 correspond to the first three flows.We introduce a 2 × 2 transition matrix M(x, y, λ), x � y; that satisfies

M ′(x, y, λ) = V (x, λ)M(x, y, λ), M(y, y, λ) = I.

The solution is given by the formula

M(x, y, λ) = exp∫ x

y

V (ξ, λ) dξ.

The matrix M(x, y, λ) is unimodular because V is traceless.The symmetry

σ1V (x, λ)σ1 = V (x, λ)

produces the same relation for the transition matrix

σ1M(x, y, λ)σ1 = M(x, y, λ). (2.3)

Another symmetry

V T (x, λ)J = −JV (x, λ),

where J = iσ2, implies

MT (x, y, λ)−1J = JM(x, y, λ). (2.4)

The quantity (λ) = 12 trace M(l, −l, λ) is called a discriminant. The for-

mula (2.3) implies (λ) = (λ) and (λ) is real for real λ. The eigenvaluesof the monodromy matrix have a name of Floquet multipliers and they are roots ofthe quadratic equation

w2 − 2 w + 1 = 0. (2.5)


The Floquet multipliers are given by the formula w = ± √ 2 − 1. The values

of λ : w(λ) = ±1 constitute the points of the periodic/antiperiodic spectrum. Thecorresponding auxiliary linear problem

(∂x − V )f = 0, f T = (f1, f2);can be written in the form of an eigenvalue problem for the self-adjoint Diracoperator

Df =[(

1 00 −1

)i∂x +

(0 −iψ

iψ 0

)]f = λ

2f .

The self-adjointness implies that points of the spectra are real.

EXAMPLE. Let ψ ≡ 0. The corresponding monodromy matrix can be easily com-puted M(x, y, λ) = e−i λ

2 σ3(x−y). We have (λ) = cos λl and double eigenvaluesat the points λ±

n = πnl. If n is even/odd, then the corresponding λ±

n belongs to theperiodic/anti-periodic spectrum.

For a generic potential the double points λ±n of the periodic/anti-periodic spec-

trum split, but they always stay real. The size of the spectral gap is determined,roughly speaking, by the corresponding Fourier coefficients of the potential. In ourconsiderations we assume that there is a finite number of g + 1 open gaps in thespectrum

· · · < λ−n−1 = λ+

n−1 < λ−n < λ+

n < · · · < λ−n+g < λ+

n+g < λ−n+g+1

= λ+n+g+1 < · · ·

These are so-called finite gap potentials which are dense among all potentials.The Floquet multipliers become single-valued on the Riemann surface:

� = {Q = (λ, w) ∈ C2: R(λ, w) = det[M(l, −l, λ) − wI ] = 0}.

The Riemann surface consists of two sheets covering the plane of the spectralparameter λ.

EXAMPLE. Let ψ ≡ 0. We have (λ) = cos λl and quadratic equation (2.5) hasthe solutions w(λ) = e±ilλ. The Riemann surface � = �+ + �− is reducible andconsists of two copies of the complex plane C that intersect each other at the pointsof the double spectrum λ±

n . Each part �+ or �− contains the corresponding infinityP+ or P−. The Floquet multipliers are single valued on �:

w(Q) = e+iλl, Q ∈ �+;w(Q) = e−iλl, Q ∈ �−.

For a finite gap potential the Riemann surface � is irreducible. There are threetypes of important points on �. These are the singular points, the points aboveλ = ∞ and the branch points which we discuss now in detail.

10 K. L. VANINSKY

• The singular points are determined by the condition∂λR(λ, w) = ∂wR(λ, w) = 0.

These are the points (λ±, ±1) of the double spectrum. At these points twosheets of the curve intersect.

• There are two nonsingular points P+ and P− above λ = ∞. At these points�

w(Q) = e+iλl(1 + O(1/λ)), Q ∈ (P+); (2.6)

w(Q) = e−iλl(1 + O(1/λ)), Q ∈ (P−). (2.7)• The branch points are specified by the condition

∂wR(λ, w) = 0.

They are different from the singular points and correspond to the simple pe-riodic/antiperiodic spectrum. We denote these points by s±

k = (λ±k , (−1)k),

k = n, . . . , n+g. There are 2(g+1) of them, each has a ramification index 2.

The desingularized curve � is biholomorphicaly equivalent to a hyperelliptic curvewith branch points at the points of the simple spectrum. We also denote the hyper-ellitic curve by �. The Riemann–Hurwitz formula for the genus of � implies

genus = R

2− n + 1,

where R is a total ramification index and n is the number of sheets. Each branchpoint has a ramification index 1 and therefore R = 2(g + 1) and n = 2. Therefore,the genus of � is g, one off the number of open gaps in the spectrum.

Let ε± be a holomorphic involution on the curve � permuting sheets

ε±: (λ, w) −→ (λ, 1/w).

The fixed points of ε± are the branch points of �. The involution ε± permutesinfinities ε± : P− → P+. Let us also define on � an antiholomorphic involution

εa: (λ, w) −→ (λ, w).

The involution εa also permutes infinities and commutes with ε±. Points of thecurve above gaps [λ−

n , λ+n ] where | (λ)| � 1 form g + 1 fixed “real” ovals of εa .

We call them a-periods.The quasimomentum p(Q) is a multivalued function on the curve �. It is intro-

duced by the formula w(Q) = eip(Q)2l . Evidently, it is defined up to πnl

, where n isan integer. The asymptotic expansion for p(Q) at infinities can be easily computed

±p(λ) = λ

2− p±

0 − p1

λ− p2

λ2. . . , Q ∈ (P±), λ = λ(Q),

where p±0 = πk±

l, k± is an integer and

p1 = 1

lH1, p2 = 1

lH2, p3 = 1

lH3, etc.

� The notation Q ∈ (P ) means that the point Q is in the vicinity of the point P .


Moreover, the function w(Q)+w(ε±Q) does not depend on the sheet and is equalto 2 (λ). Thus (λ(Q)) = cosh ip(Q)2l and the formula

dp = ± 1

i2ld cosh−1 (λ) = ± 1

i2l

•(λ) dλ√ 2 − 1

implies that differential dp is of the second kind with double poles at the infinities:±dp = d(λ

2 + O(1)). The same formula implies that the differential dp is purecomplex on the real ovals. At the same time, the condition w(s−

k ) = w(s+k ) requires

the increment p(s+k ) − p(s−

k ) to be real. Therefore, dp has zero a-periods∫ak

dp = 0.

Since the Floquet multiplies are single-valued on � for the b-periods we have∫bk

dp = πnbk

l, nbk

∈ Z, k = 1, . . . , g. (2.8)

These are so-called periodicity conditions [21].The Floquet solution is the vector-function

e(x, Q) =[

e1(x, Q)

e2(x, Q)

]

which is a solution of the auxiliary spectral problem e′ = V e with the property

e(x + 2l, Q) = M(l, −l, λ)e(x, Q) = w(Q)e(x, Q) (2.9)

and normalized by the condition

e1(−l, Q) + e2(−l, Q) = 1. (2.10)

Remark. If f (x, λ) is a solution of the auxiliary problem

(∂x − V (x, λ))f = 0

corresponding to λ, then f = σ1f is a solution of (∂x − V (x, λ))f = 0 corre-sponding to λ.

EXAMPLE. Let ψ = 0. The Floquet solution is given by the formula

e(x, Q) = e+i λ2 (x+l)e0 = e+i λ

2 (x+l)

[01

], Q ∈ �+,

e(x, Q) = e−i λ2 (x+l)e0 = e−i λ

2 (x+l)

[10

], Q ∈ �−.

It has no poles in the affine part of the curve.

For a general finite gap potential the situation is more complicated.

12 K. L. VANINSKY

LEMMA 2.1. The Floquet solution satisfies the identity

e(x, εaQ) = σ1e(x, Q).

The Floquet solution e(x, Q) has poles common for both components at the points

γ1, γ2, . . . , γg+1.

Projections of poles µk = λ(γk) are real. Each γk lies on the real oval above thecorresponding open gap [λ−

k , λ+k ]. Each component ei(x, Q) has g + 1 zeros

σ i1(x), σ i

2(x), . . . , σ ig+1(x); i = 1, 2.

These zeros depend on the parameter x. In the vicinity of infinities the functione(x, Q) has the asymptotics

e(x, Q) = e±i λ2 (x+l)[e0/e0 + o(1)], Q ∈ (P±).

Before proceeding to the proof of the lemma we note that the differential equa-tion for the monodromy matrix

M ′(x, y, λ) =[− iλ

2σ3 + Y0

]M(x, y, λ), M(y, y, λ) = I,

multiplied (gauged) on the left and right by the matrices

C =(

1 1i −i

)and C−1 = 1

2

(1 −i

1 i

)

transforms into

MR(x, y, λ) =[− iλ

2σ2 + Y R

0

]MR(x, y, λ), MR(y, y, λ) = I,

with

Y R

0 =(

q p

p −q

), ψ = q + ip.

This is a real version of the eigenvalue problem which is more convenient insome situations, see [19]. The Floquet solution eR(x, Q) corresponding to the realversion of the eigenvalue problem is related to e(x, Q) by the formula

eR(x, Q) = Ce(x, Q).

Therefore the result of the lemma for e(x, Q) follows from the corresponding resultfor eR(x, Q) given in [19]. We prefer to give a direct proof, though the gaugetransformation is behind all arguments.

Proof. The proof is based on the explicit formula for the Floquet solution. Let

M(x, −l, λ) =[

m11 m12

m21 m22

](x, −l, λ),


and M11, M12, etc., be the elements of the matrix M(l, −l, λ). The Floquet solutione(x, Q) is given by the formula

e(x, Q) = A(Q)

[m11

m21

](x, −l, λ) + (1 − A(Q))

[m12

m22

](x, −l, λ), (2.11)

where λ = λ(Q) and the coefficient A(Q) is

A(Q) = M12

M12 − M11 + w(Q)or A(Q) = w(Q) − M22

M21 − M22 + w(Q). (2.12)

To prove the formula note that the Floquet solution is a linear combination ofcolumns of the monodromy matrix M(x, −l, λ):

e(x, Q) = A(Q)

[m11

m21

](x, −l, λ) + A′(Q)

[m12

m22

](x, −l, λ), λ = λ(Q).

The normalization condition (2.10) implies A′(Q) = 1 − A(Q). At the same timethe Floquet solution is an eigenvector of the monodromy matrix

M(l, −l, λ)

[A(Q)

1 − A(Q)

]= w(Q)

[A(Q)

1 − A(Q)

].

This leads to two equations

M11A(Q) + M12(1 − A(Q)) = w(Q)A(Q),

or

M21A(Q) + M22(1 − A(Q)) = w(Q)(1 − A(Q)).

Each equation implies the corresponding formula for A(Q).The formulas (2.3) and (2.12) imply

1 − A(εaQ) = A(Q).

This and (2.3), (2.11) imply the stated identity for the Floquet solution.The relation MR = CMC−1 implies

MR =[

MR

11 MR

12

MR

21 MR

22

]

= 1

2

[M11 + M12 + M21 + M22 i(M12 + M22 − M11 − M21)

i(M11 + M12 − M21 − M22) M11 + M22 − M12 − M21

].

Due to (2.3) MR(λ) is real for real λ. Consider the function MR

12(λ) and look atthe roots µn : MR

12(µn) = 0. For ψ ≡ 0 we have MR

12(λ) = − sin λl2 with roots

at the points µn = 2πnl

, n ∈ Z. When we add the potential the roots µn movebut stay real. They are caught by open gaps or match double periodic/antiperiodicspectrum. Indeed at µn the matrix MR is lower triangular and real entries MR

11 and

14 K. L. VANINSKY

MR

22 coincide with Floquet multipliers. Since MR

11MR

22(µn) = 1 we have | (µn)| =12 |MR

11 + MR

22| � 1.The points of the divisor γk ∈ � lie above the points µk on the sheet with

w(Q) = MR

22(µn). At these points the denominator in (2.12)

M12 − M11 + w(Q) or M21 − M22 + w(Q)

vanishes. Indeed from MR

12(µn) = 0 we have

M12 − M11(µn) = M21 − M22(µn)

and w(Q) = MR

22(µn) = M11 − M12(µn) = M22 − M21(µn). MoreoverM22 �= MR

22(µn). These produce a pole of the Floquet solution when µn lies inthe open gap. When µn is caught by the periodic/antiperiodic spectrum the matrixM(l, −l, µn) = ±I and the zero of denominator is annihilated by the zero ofnumerator in (2.12).

The asimptotics of the Floquet solution follows from the formula (2.11) and

M(x, y, λ) = e−i λ2 σ3(x−y) + o(1), when λ → ∞. �

The Floquet solution e(x, Q) near infinities can be expanded into the asymptoticseries

e(x, Q) = e+i λ2 (x+l)

∞∑s=0

es(x)λ−s = e+i λ2 (x+l)

∞∑s=0

[bs

ds

]λ−s, Q ∈ (P+),

e(x, Q) = e−i λ2 (x+l)

∞∑s=0

es(x)λ−s = e−i λ2 (x+l)

∞∑s=0

[ds

bs

]λ−s, Q ∈ (P−),

and b0 = 0, d0 = 1. The coefficients bs , ds can be computed from the relation

−[

b′s

d ′s

]+ Y0

[bs

ds

]= i

2(I + σ3)

[bs+1

ds+1

], s = 0, 1, . . . , (2.13)

due to the diagonal form of the matrix

i

2(I + σ3) =

[i 00 0

].

Indeed, relation (2.13) leads to the identities

−b′s + ψds = ibs+1, (2.14)

and

−d ′s + ψbs = 0. (2.15)

These are supplemented by the boundary condition

bs(x) + ds(x)|x=−l = 0, s � 1. (2.16)


When s = 0 using b0 = 0, d0 = 1 from (2.14) we obtain

b1 = −iψ(x).

The identities (2.15) and (2.16) imply

d1 = iψ0 − i

∫ x

−l

|ψ |2 dx ′, ψ0 = ψ(−l).

Similar, we compute

b2 = ψ′ + ψ ψ0 − ψ

∫ x

−l

|ψ |2 dx ′,

d2 = −ψ′0 − ψ

20 +

∫ x

−l

[ψψ

′ + |ψ |2ψ0 − |ψ |2∫ x′

−l

|ψ |2 dx ′′]

dx ′.

These formulae will be used for explicit computation of symplectic forms.The Floquet solution e(x, Q) satisfies the identity

[J∂x − JV ]e(x, Q) = 0, J = iσ2,

which is just another way to write the spectral problem. Let us define the dualFloquet solution e+(x, Q) = [e1+(x, Q), e2+(x, Q)] at the point Q as

e+(x, Q) = e(x, ε±Q)T .

It can be verified by a direct computation that the dual Floquet solution e+(x, Q)

satisfies�

e+(x, Q)[J∂x − JV ] = 0.

The fact that the Wronskian e+(x, Q)J e(x, Q) does not depend on x can be veri-fied by differentiation. Introducing the function

�(Q) = e+(x, Q)J e(x, Q) = 〈e+(x, Q)J e(x, Q)〉= 1

2l

∫ l

−l

e+(x, Q)J e(x, Q) dx,

we define another dual Floquet solution e∗(x, Q) by the formula

e∗(x, Q) = e+(x, Q)

�(Q).

Evidently, e∗(x, Q)J e(x, Q) = 1. The symmetry (2.4) produces an analog ofmonodromy property (2.9) for the function e∗:

e∗(x + 2l, Q) = e∗(x, Q)JM−1(l, −l, λ)J−1 = w−1(Q)e∗(x, Q). (2.17)

� The action of the differential operator D = ∑kj=0 ωj ∂j on the row vector f + is defined as

f +D = ∑kj=0(−∂)j (f +ωj ).

16 K. L. VANINSKY

LEMMA 2.2. The function e∗(x, Q) has simple poles at the branch points s±k . It

has fixed zeros at γ1, . . . , γg+1. The other zeros for each component of the vectorfunction e∗ lie on every real oval and depend on the parameter x. The function e∗has the asymptotics at infinities

e∗(x, Q) = ±e∓ iλ2 (x+l)[eT

0 /eT0 + o(1)], Q ∈ (P+/P−).

Proof. The function �(Q) is meromorphic with 2(g + 1) poles on both sheetsabove points µn lying in open gaps and 2(g + 1) zeros at the branching points s±

k .At infinities it has the asymptotics

�(Q) = ±1 + o(1), Q ∈ (P±).

Now it is easy to prove properties of the function e∗(x, Q). It has poles at the branchpoints s±

k which arise from zeros of �(Q). It has zeros at γ1, . . . , γg+1, the poles of�(Q). Other poles of �(Q) are annihilated by the poles of e+(x, Q). Other g + 1zeros of e∗(x, Q) which depend on x are produced by the corresponding zeros ofe+(x, Q).

The asymptotics follows from the asymptotics for e(x, Q) and �(Q). �Consider periodic variations of the matrix V (x, λ) : V = V +δV . Then p(Q) =

p(Q) + δp(Q) + . . . We need a standard formula connecting the variations δp(Q)

and δV .

LEMMA 2.3. The following identity holds

iδp(Q) = 〈e∗(x, Q)JδV e(x, Q)〉.Proof. Let e(x, Q) be a Floquet solution corresponding to the deformed poten-

tial V . From the definition

e+(x, Q)([J∂x − J V ]e(x, Q)) = 0

and

(e+(x, Q)[J∂x − JV ])e(x, Q) = 0.

Subtracting one identity from another, we have

e+(J ∂x e) − (e+J∂x)e = e+J V e − e+JV e = e+JδV e.

Integrating both sides, we have∫ +l

−l

e+J e′ + e+′J e dx = e+J e|+l

−l .

Using the identities

e+(l, Q) = e−ip(Q)2le+(−l, Q)


and

e(l, Q) = eip(Q)2l e+(−l, Q)

for the LHS, we have

(eip(Q)2le−ip(Q)2l − 1)e+(x, Q)J e(x, Q) = iδp(Q)2l�(Q)++ lower order terms.

The RHS is equal to

2l〈e+(x, Q)JδV e(x, Q)〉 + lower order terms.

Collecting leading terms, we obtain the stated identity. �Consider a real hyperelliptic spectral curve � of finite genus corresponding

to some periodic potential ψ . Let us introduce the Baker–Akhiezer functione(τ, x, t, Q) which depends on three parameters (times) τ , x and t and has theasymptotics at infinities

e(τ, x, t, Q) = e±i(− 12 τ+ λ

2 x− λ2

2t) × [e0/e0 + o(1)], Q ∈ (P+/P−).

The BA function has poles at the points γ ’s, located on the real ovals. These prop-erties define the BA function uniquely. The BA function can be written explicitly interms of theta-functions of the curve � [10]. The BA function has Bloch propertyin x-variable e(τ, l, t, Q) = w(Q)e(τ, −l, t, Q) and satisfies the identities

[J∂τ − JV1(τ, x, t)]e(τ, x, t, Q) = 0,

[J∂x − JV2(τ, x, t)]e(τ, x, t, Q) = 0,

[J∂t − JV3(τ, x, t)]e(τ, x, t, Q) = 0.

The three matrices V1, V2 and V3 are given at the beginning of this section.Let us define the dual BA function e+(τ, x, t, Q) at the point Q as

e+(τ, x, t, Q) ≡ e(τ, x, t, ε±Q)T .

The identity w(Q)w(ε±Q) = 1 implies e+(τ, l, t, Q) = w(Q)−1e+(τ, −l, t, Q).The dual BA function e+(τ, x, t, Q) satisfies dual identities e+(τ, x, t, Q)[J∂τ −JV1] = 0, etc.

2.2. SYMPLECTIC STRUCTURES

We assumed in the previous section that the phase space consists of smooth 2l-periodic functions ψ(x). Instead we can change the language and think about thephase space as a space of operators ∂x − V2 with 2l-periodic potential. The flowsof the NLS hierarchy act on this space as well they act on the space of functions ψ .All notions of differential geometry can be applied to this space of operators withobvious conventions.

18 K. L. VANINSKY

LEMMA 2.4. The formula

ω0 =∑P±

res〈e∗JδV ∧ δe〉 dλ,

defines a closed 2-form ω0 on the space of operators ∂x−V2 with periodic potential.The flows etXm , m = 1, 2, . . . on the space of operators defined by the formula

[∂τm− Vm, ∂x − V2] = 0,

are Hamiltonian with the symplectic structure ω0 and the Hamiltonian functionHm (up to unessential constant factor).

Remark. The formula

ωn =∑P±

res λn〈e∗JδV ∧ δe〉 dλ, n = 0, 1, . . . ,

defines a closed 2-form ωn on the space of operators ∂x −V2 with periodic potentialwhich satisfy the constrains Hk = const , k = 1, . . . , n; see for details [13].

Before proceeding to the proof of the lemma, we compute the first two sym-plectic structures using the formula

ωn =∑P±

resλn

�(Q)〈e+JδV ∧ δe〉 dλ, n = 1, 2.

The result is

ω0 = 2i〈δψ ∧ δψ〉, (2.18)

ω1 = ω1 = 〈δψ ∧ δψ′ + δψ ∧ δψ ′ + 2δ∂−1|ψ |2 ∧ δ|ψ |2〉 (2.19)

subject to the constraint H1 = const. We present the computation divided in smallsteps.

Step 1. Identity �(τ±Q) = −�(Q) implies

1

�(Q)= φ0 + φ1

λ+ φ2

λ2+ · · · , Q ∈ (P+),

1

�(Q)= −φ0 − φ1

λ− φ2

λ2− · · · , Q ∈ (P−).

Using definition of e(x, Q) and e+(x, Q) from previous section, we have

φ0 = 1,

φ1 = −〈d1 + d1〉.Step 2. Near P+ we obtain

〈e+JδV ∧ δe〉 = s1

λ+ s2

λ2+ s3

λ3+ · · · ,


where

s1 = 〈eT0 JδV ∧ δe1〉,

s2 = 〈eT0 JδV ∧ δe2〉 + 〈eT

1 JδV ∧ δe1〉.Similar, at P−, we obtain

〈e+JδV ∧ δe〉 = − s1

λ− s2

λ2− s3

λ3− · · · .

Using the expansion for �(Q) from Step 1, we derive

ω0 = φ0(s1 + s1),

ω1 = φ0(s2 + s2) + φ1(s1 + s1).

Step 3. Computing s1, we have

s1 = 〈eT0 JδV ∧ δe1〉 = 〈δψ ∧ δb1〉.

Using the formula b1 = −iψ , we obtain s1 = −i〈δψ ∧ δψ〉 and (2.18).

Step 4. The first term in the formula for s2 produces

〈eT0 JδV ∧ δe2〉 = 〈δψ ∧ δb2〉.

Using recurrence relation (2.14) b2 = ib′1 − iψd1, we obtain

〈δψ ∧ iδb′1 − iδψd1 − iψδd1〉.

The second term in the formula for s2 produces

〈eT1 JδV ∧ δe1〉 = 〈−b1δψ ∧ δd1 + d1δψ ∧ δb1〉.

Finally,

s2 = 〈δψ ∧ δψ′ − i(d1 + d1)δψ ∧ δψ − iδ|ψ |2 ∧ δd1〉.

Using the formula for s1,

ω1 =< δψ ∧ δψ′ + δψ ∧ δψ ′ − iδ|ψ |2 ∧ δ(d1 − d1)〉.

The constraint H1 = const implies 〈δ|ψ |2〉 = 0 and using the explicit formula ford1, we obtain (2.19).

Proof. Closeness of the form ω0 follows from the result of next lemma or fromexplicit formula (2.18). For the second statement we present a complete proof onlyfor m = 0. Higher flows can be treated similarly. In order to prove that the firstflow is Hamiltonian one has to establish [1],

i∂tω0 = −δ2H1,

20 K. L. VANINSKY

where i∂tis the contraction operator produced by the vector field X1. Using time-

dependent BA functions i∂tδe = e•, i∂t

δV = V • we have

i∂tω0 =

∑P±

res〈e∗JV •δe〉 dλ − res〈e∗JδV e•〉 dλ.

Let us compute the residue at P+. From the computation preceding the proof

1

�(Q)= φ0 + φ1

λ+ · · · , Q ∈ (P+),

1

�(Q)= −φ0 − φ1

λ− · · · , Q ∈ (P−).

Then, using V • = [ i2σ3, V ] = iσ3V , we obtain

res〈e∗JV •δe〉 dλ = iφ0〈eT0 Jσ3Y0δe1〉.

Similarly, using e• = i2σ3e we have:

res〈e∗JδV e•〉 dλ = i2φ1〈eT0 JδY0σ3e0〉 +

+ i2φ0[〈eT0 JδY0σ3e1〉 + 〈eT

1 JδY0σ3e0〉].The first term vanishes and

i∂tres〈e∗JδV ∧ δe〉 dλ = iφ0〈eT

0 Jσ3Y0δe1〉 −− i

2φ0[〈eT

0 JδY0σ3e1〉 + 〈eT1 JδY0σ3e0〉].

Similarly at P−

i∂tres〈e∗JδV ∧ δe〉 dλ = −iφ0〈eT

0 Jσ3Y0δe1〉 ++ i

2φ0[〈eT

0 JδY0σ3e1〉 + 〈eT1 JδY0σ3e0〉].

Finally, we obtain

i∂tω0 = iφ0[〈eT

0 Jσ3Y0δe1〉 − 〈eT0 Jσ3Y0δe1〉] = −δ2H1. �

2.3. DARBOUX COORDINATES

The formulas (2.18) and (2.19) give examples of symplectic forms. All these formscan be put in the Darboux form in the coordinates associated with poles of theBaker–Akhiezer function.


LEMMA 2.5. The formula

ξ0(Q) = 〈e∗JδV ∧ δe〉 dλ

defines meromorphic in Q differential 2-form on � with poles at γ1, . . . , γg+1 andP+, P−. The symplectic 2-form defined by the formula

ω0 =∑P±

res ξ0(Q)

can be written as

ω0 = 2

i

g+1∑k=1

δp(γk) ∧ δλ(γk).

Remark 1. The meaning of the right-hand side of this formula is the following.The curve � (or its cover �) is equipped with two meromorphic functions λ(Q) andp(Q). Their variation δp(Q) and δλ(Q) at the points of the divisor is computedfor variation of the potential ψ(x), ψ(x); −l � x � l. The RHS of the formula isthe sum of an exterior products of these variations.

Remark 2. In fact for a general smooth potential the divisor λ(γk) and val-ues of the quasimomentum cosh−1 (λ(γk)) with suitably chosen sign (= sheet)determine the potential. In other words they are global coordinates on the phasespace. First note that the discriminant can be reconstructed from this data usingShannon interpolation, see [19]. Thus the curve � is known. The potential can beeffectively recovered from the divisor via trace formulas, see [18].

Proof. Note δV = δY0 does not depend on λ. Essential singularity of the Floquetsolutions at P± cancels out, and infinities are simple poles for the form ξ0(Q). Inthe finite part of the curve ξ0(Q) has two sets of poles. One is the poles γ1, . . . , γg+1

of the Baker–Akhiezer function. Another is the branch points of the curve �. Bythe Cauchy theorem

∑P±

res ξ0(Q) = −∑γk

res ξ0(Q) −∑sk

res ξ0(Q).

Let us compute contribution of the first set of poles. Near γk we have

e = res e

λ − λ(γk)+ O(1).

Therefore,

δe = res e

(λ − λ(γk))2δλ(γk) + O

(1

λ − λ(γk)

),

22 K. L. VANINSKY

and

δe = e

λ − λ(γk)δλ(γk) + O

(1

λ − λ(γk)

). (2.20)

Note that e∗(x, γk) ≡ 0 and from Lemma 2.3 we obtain

resγkξ0(Q) = 〈e∗JδV e〉(γk) ∧ δλ(γk)resγk

[dλ

λ − λ(γk)

]= iδp(γk) ∧ δλ(γk).

Now consider branch points of the curve. They produce a nontrivial contribu-tion, though the pole of e∗ at the branch point is annihilated by the zero of thedifferential dλ. Nevertheless, the variation δe(x, Q) has a simple pole at sk. First,let us make a general remark.

Consider, the variation of a function f (Q, ψ, ψ) under variation of the potentialψ(x), ψ(x), −l � x � l; taken for Q in the vicinity of the branch point sk anda fixed value of λ. Such variation will have a pole at the branch point itself. Atthe branch point λ fails to be a local parameter, but w is fine due to the fact∂λR(λ, w)|sk �= 0. Now, consider a function f (Q, ψ, ψ) = f (w, ψ, ψ), and defineits variation δ0 for a fixed value of w. Then,

δf = δ0f + df

dwδw.

Take, for example, f (Q) = λ(Q), then

0 = δ0λ + dλ

dwδw. (2.21)

Therefore, for a general f we have

δf = δ0f − df

dλδ0λ = −df

dλδ0λ + O(1). (2.22)

The zero of the differential dλ at the branch point produces the pole of δf .We can proceed to the computation of the residues of ξ0(Q) at sk. In the local

parameter (λ − λ(sk))1/2 ∼ w − 1, λ = λ(Q):

δe(x, Q) = −e1(x)

2

δλ(sk)

(λ − λ(sk))1/2+ · · · ,

where

e(x, Q) = e0(x) + e1(x)(λ − λ(sk))1/2 + · · · , Q ∈ (sk).

Similarly,

de(x, Q) = e1(x)

2

dλ

(λ − λ(sk))1/2+ · · · ,


and we have

δe(x, Q) = −de(x, Q)

dλδλ(sk) + O(1).

The leading term is the same as in general formula (2.22). Therefore, (2.21) implies

ressk ξ0(Q) = −ressk [〈e∗JδV de〉] ∧ δλ(sk) = ressk

[〈e∗JδV de〉 ∧ dλδw

dw

].

Now using

e∗(x, Q) = e∗(l, Q)MT (l, x, λ)−1,

de(x, Q) = dM(x, −l, λ)e(−l, Q) + M(x, −l, λ) de(−l, Q)

with the help dM(x, y, λ)|sk = 0 we obtain

ressk ξ0(Q)

= ressk

[e∗(l, Q)〈MT (l, x, λ)−1JδV M(x, −l, λ)〉 de(−l, Q) ∧ dλδw

dw

].

Symmetry (2.4) of the monodromy matrix implies

〈MT (l, x, λ)−1JδV M(x, −l, λ)〉 = 1

2lJ δM(l, −l, λ),

and using skew-symmetry of the wedge product

ressk ξ0(Q) = 1

2lressk

[e∗(l, Q)J δM(l, −l, λ) de(−l, Q) ∧ dλδw

dw

]

= 1

2lressk

[e∗(l, Q)J (δM(l, −l, λ) − δw) de(−l, Q) ∧ dλδw

dw

].

Identities (2.9) and (2.17) imply

e∗J (δM − δw) = δe∗J (w − M),

J (w − M) de = J (dM − dw)e.

Therefore,

ressk ξ0(Q) = 1

2lressk

[δe∗(l)J (dM(l, −l, λ) − dw)e(−l) ∧ dλδw

dw

].

Since e∗(l)J e(−l) = w−1, we have

ressk ξ0(Q) = 1

2lressk [e∗(l)J δe(−l) ∧ δw dλ].

The one form

e∗(l, Q)J δe(−l, Q) ∧ δw(Q) dλ(Q)

24 K. L. VANINSKY

is holomorphic (in the parameter λ) outside of the poles γk and the branch points.At infinity the essential singularity cancels out and due to (2.6)–(2.7), Lemmas 2.1and 2.2

e∗(l, Q)J δe(−l, Q) ∧ δw(Q) = o

(1

λ

).

This implies

resP±[e∗(l, Q)J δe(−l, Q) ∧ δw(Q) dλ(Q)] = 0.

By the Cauchy theorem,∑sk

ressk [e∗(l)J δe(−l) ∧ dλδw] = −∑γk

resγk[e∗(l)J δe(−l) ∧ dλδw].

Therefore, using e∗(l) = w−1e∗(−l), we have

∑sk

ressk ξ0(Q) = − 1

2l

∑γk

resγk[e∗(l)J δe(−l) ∧ dλδw]

= − 1

2l

∑γk

Rγk

[e∗(−l)J δe(−l) ∧ dλ

δw

w

].

Using the formula (2.20) we finally obtain∑sk

ressk ξ0(Q) =∑γk

iδp(γk) ∧ resγk[e∗(−l)J δe(−l) dλ]

=∑γk

iδp(γk) ∧ δλ(γk)resγk

[dλ

λ − λ(γk)

]

=∑γk

iδp(γk) ∧ δλ(γk). �

Remark. For the higher symplectic structures an analogous result holds

ωn = 2

i

g+1∑k=1

λnδp(γk) ∧ δλ(γk), n = 0, 1, . . . ;

subject to the constrains Hk = const , k = 1, . . . , n.

2.4. ACTION-ANGLE VARIABLES

Here we describe briefly another system of Darboux coordinates. We refer to thepaper [19] for details.


The actions Ik, k = 1, . . . , g + 1, are defined by the formula

Ik = 1

4π

∫ak

p(λ) dλ.

In the formula above the multivalued function p(λ) is normalized such thatp(sk) = 0, k = 1, . . . , g + 1. The angles θk, n = 1, . . . , g + 1, are

θk =g+1∑n=1

∫ γn

sn

αk.

The differentials αk, n = 1, . . . , g + 1, are of the third kind with poles at theinfinities P± normalized in such a way that�

∫ak

αn = 2πδkn.

As it is proved in [19] by a direct computation

{θn, Ik} = δkn.

All other brackets vanish

{In, Ik} = 0, {θn, θk} = 0.

These formulas imply the identity for symplectic forms.

ω0 = 2g+1∑n=1

δIn ∧ δθn.

We conclude this section with a few remarks.

Remark 1. Another way to prove the identity for symplectic forms withoutemploying the Poisson bracket is found by Krichever [11].

Remark 2. McKean [17], proved various identities for 1-forms.

Remark 3. In the finite gap case the curve � is specified by 2(g + 1) branchpoints. The actions Ik, k = 1, . . . , g+1, together with the g other periods (see (2.8))of the differential dp and the constant p+

0 determine the curve. This fact is due toKrichever [12] (see also [2]). In the infinite gap case it is shown in [19] that I − θ ’sare global coordinates on the phase space for a general square integrable potential.This property holds for a finite gap potential as well.

� δkn is Kronecker delta.

26 K. L. VANINSKY

3. The Scattering Case

3.1. JOST SOLUTIONS

In the next sections we consider the scattering problem for the Dirac operator on theentire line with rapidly decaying potential. The Riemann–Hilbert approach to thescattering theory for canonical systems with summable potential was constructedby M. G. Krein and P. E. Melik-Adamian [8, 9, 16]. This approach was used manytimes in soliton theory [4].

To simplify the estimates we assume that the potential ψ is from the Schwartz’sspace S(R) of complex rapidly decreasing infinitely differentiable functions on theline such that

supx

|(1 + x2)nψ(m)(x)| < ∞, m, n = 0, 1, . . . .

Let us introduce the reduced transition matrix T (x, y, λ), x � y; by the formula

T (x, y, λ) = E−1

(λx

2

)M(x, y, λ)E−1

(−λy

2

), (3.1)

where E(λx2 ) = exp(− iλx

2 σ3) is a solution of the free equation (ψ ≡ 0). The matrixT (x, y, λ) solves the equation

T ′(x, y, λ) = Y0(x)E(λx)T (x, y, λ), T (y, y, λ) = I.

The spectral parameter enters multiplicatively into the RHS of the differentialequation. The solution is given by the formula

T (x, y, λ) = exp∫ x

y

Y0(ξ)E(λξ) dξ. (3.2)

The symmetry of the matrix Y0 : σ1Y0(x)σ1 = Y0(x) is inherited by unimodularmatrix T:

σ1T (x, y, λ)σ1 = T (x, y, λ).

For real λ the formula (3.2) and the rapid decay of the potential imply an existenceof the limit

T (λ) = lim T (x, y, λ) =(

a(λ) b(λ)

b(λ) a(λ)

), when y → −∞ and x → +∞

and |a(λ)|2 − |b(λ)|2 = 1. When the potential ψ ∈ S(R), we have b(λ) ∈ S(R).We introduce Jost solutions J±(x, λ) as a matrix solutions of the differential

equation

J ′±(x, λ) = V (x, λ)J±(x, λ),

J±(x, λ) = E

(λx

2

)+ o(1), when x → ±∞.


An existence and analytic p roperties of the Jost solutions follow from the integralrepresentations

J+(x, λ) = E

(xλ

2

)+

∫ +∞

x

�+(x, ξ)E

(λξ

2

)dξ,

J−(x, λ) = E

(xλ

2

)+

∫ x

−∞�−(x, ξ)E

(λξ

2

)dξ.

The kernels �± are unique and infinitely smooth in both variables. Introducing thenotation J± = [j (1)

± , j(2)± ] we see from the integral representations that j

(1)− (x, λ),

j(2)+ (x, λ) are analytic in λ in the upper half-plane and continuous up to the bound-

ary. Also, the columns j(2)− (x, λ), j (1)

+ (x, λ) are analytic in the lower half-plane andcontinuous up to the boundary.

Now we describe analytic properties of the coefficient a(λ) of the matrix T (λ).The monodromy matrix M(x, y, λ) can be written in the form

M(x, y, λ) = J+(x)J−1+ (y) = J−(x)J−1

− (y).

Therefore,

J−1+ (x)M(x, y, λ)J−(y) = J−1

+ (y)J−(y) = J−1+ (x)J−(x).

The variables x and y separate and the above expression does not depend on x ory at all. By passing to the limit with x → +∞, y → −∞ we have

T (λ) = J−1+ (y)J−(y) = J−1

+ (x)J−(x).

Therefore,

a(λ) = j(1)−

T(λ)Jj

(2)+ (λ).

The properties of Jost solutions imply that

• a(λ) is analytic in the upper half-plane and continuous up to the boundary;• a(λ) is root-free;• |a(λ)| � 1 and |a(λ)|2 − 1 ∈ S(R) for λ real, a(λ) = 1 + o(1) as |λ| → ∞.

This coefficient will be used to construct the scattering curve �∞.Let p∞(λ) be such that a(λ) = exp(−i2p∞(λ)) for λ in the upper half-plane.

The quantity p∞(λ) in analogous to the quasimomentum studied in the periodiccase, see [26].� From the properties of a(λ) it follows that p∞(λ) is analytic in theupper half-plane and continuous up to the boundary; �p∞(λ) � 0 for �λ � 0;p∞(λ) = o(1) for |λ| → ∞; for real λ, the density of the measure dµ∞(λ) =�p∞(λ) dλ belongs to S(R). The function p∞(λ) can be written in the form

p∞(λ) = 1

π

∫dµ∞(t)

t − λ.

� The spectral parameter λ in the paper [26] has different scaling and should be replaced by λ/2.

28 K. L. VANINSKY

Expanding the denominator in inverse powers of λ, we obtain:

p∞(λ) = −∞∑

k=0

1

λk+1

1

π

∫ +∞

−∞tk dµ∞(t) = −H1

λ− H2

λ2− H3

λ3+ · · · , (3.3)

where H1, H2 and H3 are the integrals introduced above with l = +∞. Theexpansion has an asymptotic character for λ : δ � arg λ � π − δ, δ > 0.

To describe the asymptotic behavior in x of the Jost solutions j(2)+ (x, λ) and

j(1)− (x, λ) we assume that λ is real and fixed. Then,

x → −∞ x → +∞j

(2)+ a(λ)f →(x, λ) − b(λ)f ←(x, λ) f →(x, λ)

j(1)− f ←(x, λ) a(λ)f ←(x, λ) + b(λ)f →(x, λ),

where

f ←(x, λ) =[

e−i λ2 x

0

], f →(x, λ) =

[0

ei λ2 x

]

are solutions of the free equation.We sketch the derivation of the asymptotics for j

(2)+ (x, λ), when x → −∞.

Let ψ be a potential that vanishes outside the segment [−L, +L]. In this caseformula (3.2) becomes

T (λ) = exp∫ L

−L

Y0(ξ)E(λξ) dξ,

and T (λ) is an entire unimodular function of λ of the form

T (λ) =[

a(λ) b(λ)

b(λ) a(λ)

].

From the definition of the matrix M:

j(2)+ (L, λ) = M(L, −L, λ)j

(2)+ (−L, λ), (3.4)

and from (3.1)

M(L, −L, λ) =[

a(λ)e−iλL b(λ)

b(λ) a(λ)eiλL

].

Obviously, j(2)+ (L, λ) = f →(L, λ) and j

(2)+ (−L, λ) = c1f →(−L, λ) +

c2f ←(−L, λ) with unknown coefficients c1 and c2. Formula (3.4) for real λ leadsto the linear system

c1b(λ) + c2a(λ) = 0,

c1a(λ) + c2b(λ) = 1.


Solving for c’s we obtain the stated formula. For a potential with noncompactsupport one has to take L sufficiently large to make the error negligible.

The Riemann surface �∞ is obtained by gluing together along the real line twocopies of the complex plane (see Figure 2). One copy we call “+” and another “−”.Each copy has an infinity P+ or P−. The point Q ∈ �∞ is determined by λ = λ(Q)

and specification of the sheet Q = (λ, ±). Let us define for the “+” copy j(x, Q)

to be j(2)+ (x, λ) if �λ > 0; and j

(2)− (x, λ) if �λ < 0. For the “−” copy we define

j(x, Q) to be j(1)− (x, λ) if �λ > 0; and j

(1)+ (x, λ) if �λ < 0. In the vicinity of P±

the function j(x, Q) has asymptotics

j(x, Q) = e±i λ2 x[j 0/j 0 + o(1)], (3.5)

where λ = λ(Q) and

j 0 =[

01

], j 0 =

[10

].

Therefore, j(x, Q) can be viewed as a BA function for the singular curve �∞.We also introduce the matrix BA function

H+(λ) = [j (1)− (λ), j

(2)+ (λ)] and H−(λ) = [j (1)

+ (λ), j(2)− (λ)]

analytic in the upper/lower half-plane respectively. They are connected by thegluing condition

H−(x, λ) = H+(x, λ)S(λ), where λ ∈ R (3.6)

and the scattering matrix S(λ)

S(λ) = 1

a

[1 b

−b 1

].

The adjoint (dual) Jost solution j+ at the point Q is defined by the formula

j+(x, Q) ≡ j(x, Q)T .

Any Jost solution satisfies [J∂x − JV ]j = 0. By analogy with the periodic caseone can prove that j+ satisfies j+[J∂x − JV ] = 0.

The matrices H++ and H+

− are defined as

H++ (λ) = σ1H

T+ (λ) =

[j

(2) T+

j(1) T−

], H+

− (λ) = σ1HT− (λ) =

[j

(2) T−

j(1) T+

].

Extending a into the lower half-plane by the formula a∗(λ) = a(λ), we define

H ∗+(λ) = − σ3

a(λ)H+

+ (λ) for �λ > 0;

30 K. L. VANINSKY

and

H ∗−(λ) = − σ3

a∗(λ)H+

− (λ) for �λ < 0.

It is easy to check that the dual gluing condition holds

H ∗−(x, λ) = S−1(λ)H ∗

+(x, λ), where λ ∈ R (3.7)

and

S−1(λ) = 1

a∗

[1 −b

b 1

].

Next two lemmas state asymptotic properties of Jost solutions which will beused in computations with symplectic forms.

LEMMA 3.1. (i) For fixed x the following formulas hold

j(2)+ (x, λ) = e+i λ

2 x

∞∑s=0

j s(x)λ−s = e+i λ2 x

∞∑s=0

[gs

ks

]λ−s,

where g0 = 0, k0 = 1, and �

j(1)− (x, λ) = e−i λ

2 x

∞∑s=0

j s(x)λ−s = e−i λ2 x

∞∑s=0

[ks

gs

]λ−s,

where g0 = 0, k0 = 1. The expansion has an asymptotic character for λ : δ �arg λ � π − δ, δ > 0.

(ii) The coefficients g1, k1 are given by the formulas

g1 = −iψ, k1 = i

∫ +∞

x

|ψ(x ′)|2 dx ′

and

g1 = iψ, k1 = i

∫ x

−∞|ψ(x ′)|2 dx ′.

Proof. (i) Using� ∂nξ �

(2)+ (x, ξ)|ξ=∞ = 0, for n = 0, 1, . . . and integrating n times

by parts,

j(2)+ (x, λ) = e

iλx2

[01

]+

∫ ∞

x

�(2)+ (x, ξ)e

iλξ2 dξ

. . . = eiλx2

[01

]− e

iλx2

iλ2

�(2)+ (x, x) + e

iλx2

( iλ2 )2

∂ξ�(2)+ (x, x) − · · · +

� Operation ˆ applied to a scalar signifies complex conjugation and reversal of infinities, seeformulas in the part (ii) below.

� � = [�(1), �(2)].


+ (−1)n eiλx2

( iλ2 )n

∂n−1ξ �

(2)+ (x, x) +

+ (−1)n 1

( iλ2 )n

∫ ∞

x

∂nξ �

(2)+ (x, ξ)e

iλξ2 dξ.

This implies the existence and asymptotic character of the expansion in the para-meter λ. The other infinity can be treated similarly.

(ii) Consider j(2)+ (x, λ) first. The differential equation j ′ = V j implies

−[

g′s

k′s

]+ Y0

[gs

ks

]= i

2(I + σ3)

[gs+1

ks+1

], s = 0, 1, . . . ,

and g0 = 0, k0 = 1.This recurrent relation leads to the identities

−g′s + ψks = igs+1,

−k′s + ψgs = 0.

For s � 1 we have the boundary condition

gs(x)|x=+∞ = ks(x)|x=+∞ = 0.

These imply the stated formulas for g1, k1.For j

(1)− (x, λ) the differential equation implies

−[

k′s

g′s

]+ Y0

[ks

gs

]= i

2(−I + σ3)

[ks+1

gs+1

], s = 0, 1, . . . ,

and k0 = 1, g0 = 0.The recurrent relation produces the identities

−g′s + ψks = −igs+1,

−k′s + ψgs = 0.

For s � 1 we have the boundary condition

gs(x)|x=−∞ = ks(x)|x=−∞ = 0.

These imply the stated formulas for g1, k1. We are done. �Remark. It is interesting to compare asymptotic expansions for j

(1)− /j

(2)+ and

e(x, Q). For the Jost solution j(1)− (x, λ) normalized at the left

j(1)− (x, λ) = e− iλx

2

([10

]+ 1

λ

[i∫ x

−∞ |ψ |2iψ

]+ · · ·

)

32 K. L. VANINSKY

and

ei λ2 le(x, Q) = e− iλx

2

([10

]+ 1

λ

[−iψ0 + i∫ x

−l|ψ |2

iψ

]+ · · ·

), Q ∈ (P−).

If ψ is compactly supported and l becomes sufficiently large, then ei λ2 le(x, Q) =

j(x, λ). For the Jost solution normalized at the right the situation is slightly differ-ent. If one defines the new e(x, Q) ≡ e(x, Q)w−1(Q) which is the Floquet solutionnormalized at the right end x = l of the interval, then for compactly supported ψ

and sufficiently large l the new e−i λ2 le(x, Q) = j

(2)+ (x, λ).

A result similar to Lemma 4.1 holds for Jost solutions analytic in the lowerhalf-plane.

LEMMA 3.2. (i) For fixed x the following formulas hold

j(1)+ (x, λ) = e−i λ

2 x

∞∑s=0

j s(x)λ−s = e−i λ2 x

∞∑s=0

[hs

fs

]λ−s,

where h0 = 1, f0 = 0, and

j(2)− (x, λ) = e+i λ

2 x

∞∑s=0

j s(x)λ−s = e+i λ2 x

∞∑s=0

[fs

hs

]λ−s,

where h0 = 1, f0 = 0. The expansion has an asymptotic character forλ : δ � arg λ � −π + δ, δ > 0.

(ii) The coefficients h1, f1 are given by the formulas

f1 = iψ, h1 = −i

∫ +∞

x

|ψ(x ′)|2 dx ′

and

f1 = −iψ, h1 = −i

∫ x

−∞|ψ(x ′)|2 dx ′.

Similar to the periodic case we will need time dependent BA functions. Theyare obtained by an elementary construction.

LEMMA 3.3 [7]. There exists the Jost solution j(τ, x, t, Q) on the curve �∞ withthree time parameters τ , x and t which satisfies the differential equations:

[∂τ − V1(τ, x, t)]j(τ, x, t, Q) = 0,

[∂x − V2(τ, x, t)]j(τ, x, t, Q) = 0,

[∂t − V3(τ, x, t)]j(τ, x, t, Q) = 0.


Proof. We consider “+” sheet and the upper half-plane where j(x, Q) =j

(2)+ (x, λ). First, we construct j(τ, x, t, Q) such that

[∂x − V2(τ, x, t)]j(τ, x, t, Q) = 0

normalized for all τ and t as

j(τ, x, t, Q) ∼ e+i λ2 x

([01

]+ o(1)

), when x → ∞.

Then, we construct j(τ, x, t, ε±Q) on the lower sheet as a solution

[∂x − V2(τ, x, t)] j(τ, x, t, ε±Q) = 0

for the same value of the spectral parameter λ = λ(Q) normalized for all τ and t

as

j(τ, x, t, ε±Q) ∼ e−i λ2 x

([10

]+ o(1)

), when x → −∞.

The solutions j(τ, x, t, Q) and j(τ, x, t, ε±Q) span the kernel of the operator[∂x − V2(τ, x, t)]. Now we introduce

j new(τ, x, t, Q) ≡ e−i 12 τ−i λ2

2 tj(τ, x, t, Q),

which is the desired solution. Evidently,

[∂x − V2(τ, x, t)]j new(τ, x, t, Q) = 0.

To prove the first identity of the statement we note, that commutativity of theoperators ∂τ − V1 and ∂x − V2 implies

[∂τ − V1(τ, x, t)]j new(τ, x, t, Q) = c1(τ, t)j(τ, x, t, Q) ++ c2(τ, t)j(τ, x, t, ε±Q).

From another side as x → +∞,

[∂τ − V1(τ, x, t)]j new(τ, x, t, Q)

=[∂τ − i

2σ3

]e−i 1

2 τ+i λ2 x−i λ2

2 t

([01

]+ o(1)

)= o(1).

Due to the linear independence of the solutions j(τ, x, t, Q) and j(τ, x, t, ε±Q)

we have c1(τ, t) = c2(τ, t) = 0.Similarly it can be proved that

[∂t − V3(τ, x, t)]j new(τ, x, t, Q) = 0.

Another sheet of �∞ can be treated the same way. We are done. �

34 K. L. VANINSKY

Remark. It is easy to see that,

j new(τ, x, t, Q) = e±i(− 12 τ+ λ

2 x− λ22 t)[j 0/j 0 + o(1)] Q ∈ (P±).

Thus the standard Jost solution with asymtotics (3.5) can be obtained from the BAfunctions if one puts τ and t equal to 0.

3.2. THE SYMPLECTIC STRUCTURES

We are ready to introduce the scattering version of the Krichever–Phong formula.The everaging is defined now as an integral over the entire line 〈•〉 = ∫ +∞

−∞ dx.

THEOREM 3.4. The formula

ω0 = trace res 12 [〈H ∗+JδV ∧ δH+〉 + 〈H ∗−JδV ∧ δH−〉] dλ (3.8)

defines a closed 2-form ω0 on the space of operators ∂x − V2 with potential fromthe Schwartz class S(R). The flows etXm, m = 1, 2, . . . on the space of operatorsdefined by the formula

[∂τm− Vm, ∂x − V2] = 0,

are Hamiltonian with respect to the 2-form ω0 with Hamiltonian function Hm

(up to a nonessential constant factor).

Remark 1. The symbol

res〈H ∗+JδV ∧ δH+〉

means the coefficient corresponding to the term 1λ

in the power series expansionnear infinity in the upper half-plane. The second term

res〈H ∗−JδV ∧ δH−〉

is defined in the same way, only the upper half-plane plane is replaced with thelower half-plane.

Remark 2. The formula

ωn = trace resλn

2[〈H ∗

+JδV ∧ δH+〉 + 〈H ∗−JδV ∧ δH−〉] dλ,

where n = 0, 1, . . . , defines a closed 2-forms ωn on the space of operators∂x − V2 with potential from the Schwartz class S(R) that satisfy the constraintsHk = const , k = 1, . . . , n. It is instructive to compute explicitly the symplecticforms for small n. The first few are given by the formulas

ω0 = 2i〈δψ ∧ δψ〉,


and

ω1 =⟨δψ ∧ δψ

′ + δψ ∧ δψ ′ + δ

[∫ x

−∞|ψ |2 −

∫ ∞

x

|ψ |2]

∧ δ|ψ |2⟩.

subject to the constrain H1 = const. The derivation employs Lemmas 3.1 and 3.2and similar to the periodic case.

Proof. Closeness of the form ω0 follows either from the explicit formula or fromthe result of the next theorem. The proof of the second statement we present for thefirst etX1 flow. The time dependent Jost solutions entering into (3.9) are constructedin Lemma 3.4. Let i∂t

be the construction operator produced by the vector field X1.We will prove i∂t

ω0 = −δ2H1. For the first term in (3.8)

trace res〈H ∗+JδV ∧ δH+〉 dλ

= res1

a〈j (1)T

− JδV ∧ δj(2)+ 〉 dλ − res

1

a〈j (2)T

+ JδV ∧ δj(1)− 〉 dλ. (3.9)

Applying the contraction operator to the first term in (3.9)

i∂tres

1

a〈j (1)T

− JδV ∧ δj(2)+ 〉 dλ

= res1

a〈j (1)T

− JV •δj (2)+ 〉 dλ − res

1

a〈j (1)T

− JδV j(2)•+ 〉 dλ.

From (3.3) we have

1

a= a0 + a1

λ+ · · · , where a0 = 1, a1 = −i

∫ +∞

−∞|ψ |2.

Using V • = [ i2σ3, V ] = iσ3V , and Lemma 3.1, we have

res1

a〈j (1)T

− JV •δj (2)+ 〉 dλ = a0〈jT

0 J iσ3V δj 1〉 = −〈ψδψ〉.

Similarly, using j • = i2σ3j , we have

res1

a〈j (1)T

− JδV j(2)•+ 〉 dλ = a1

⟨j

T

0 JδVi

2σ3j 0

⟩+

+ a0

[⟨j

T

0 JδVi

2σ3j 1

⟩+

⟨j

T

1 JδVi

2σ3j 0

⟩].

The first term vanishes, the second produces

= 12〈ψδψ − ψδψ〉.

Finally,

i∂tres

1

a〈j (1)T

− JδV ∧ δj(2)+ 〉 dλ = −δH1.

36 K. L. VANINSKY

The second term in formula (3.9) can be treated similarly. Therefore,

i∂ttrace res〈H ∗

+JδV ∧ δH+〉 = −2δH1.

The second term in formula (3.8) produces the same result. The proof is finished. �

3.3. ACTION-ANGLE VARIABLES

In response to infinitesimal deformations of the matrix V = V + δV the matrixT (λ) changes according to the rule: T (λ) = T (λ) + δT (λ) + · · ·. The next resultis similar to Lemma 2.3 of the periodic case.

LEMMA 3.5. The following formula holds

〈H++ JδV H+〉 =

[ −δa bδa − aδb

aδb − bδa −δa

],

with averaging defined as

〈H++ JδV H+〉 =

∫ +∞

−∞H+

+ (x, λ)J δV (x)H+(x, λ) dx.

Proof. Let us assume, first, that ψ has compact support. We denote by V , T andj deformed matrices V, T and the Jost solution j . We will derive the expressionfor 〈j (2) T

+ JδV j(1)+ 〉 in the left-upper corner. First, we obtain the formula

〈j+JδV j〉 + lower order terms = j+J j |+L−L. (3.10)

Indeed,

j+([J∂x − J V ]j) = 0,

(j+[J∂x − JV ])j = 0.

Subtracting one identity from another we have

j+JδV j + lower order terms = j+(J ∂ j) − (j+J∂)j .

Integrating the RHS in x variable we obtain∫ +L

−L

[j+J j′ + j+′J j ] dx = j+J j |+L

−L.

This implies (3.10). Now using formulas for the asymptotics of j(2)+ and j

(1)− for

the RHS of (3.10), we have

j(2)T+ J j

(1)

− |+L−L = f T

→J [af ← + bf →] |+L − [af T→ − bf T

←]Jf ← |−L

= af T→Jf ← − af T

→Jf ← = a − a = −δa ++ lower order terms.


Collecting terms of the same order, we obtain the result. The case of a potentialwith noncompact support can be considered using approximation arguments. Forother entries the arguments are the same. Lemma is proved. �THEOREM 3.6. The following formulas hold

ω0 = 1

πi

∫ +∞


|a(λ)|2 dλ.

Proof. By the Cauchy integral formula

12 trace res〈H ∗+JδV ∧ δH+〉 = − 1

2πi

∫ +∞

−∞trace〈H ∗

+JδV ∧ δH+〉 dλ,

and

12 trace res〈H ∗−JδV ∧ δH−〉 = 1

2πi

∫ +∞

−∞trace〈H ∗

−JδV ∧ δH−〉 dλ.

Taking sum

ω0 = 1

2πi

∫ +∞

−∞trace〈H ∗

−JδV ∧ δH−〉 dλ −

− 1

2πi

∫ +∞

−∞trace〈H ∗

+JδV ∧ δH+〉 dλ.

Using (3.6) and (3.7)

δH− = δH+S + H+δS,

we obtain

ω0 = 1

2πi

∫ +∞

−∞trace S−1〈H ∗

+JδV ∧ δH+〉S dλ +

+∫ +∞

−∞trace S−1〈H ∗

+JδV H+〉 ∧ δS dλ −

− 1

2πi

∫ +∞

−∞trace〈H ∗

+JδV ∧ δH+〉 dλ

=∫ +∞

−∞trace〈H ∗

+JδV H+〉 ∧ δS S−1 dλ.

Now, applying the result of Lemma 3.5, we have

ω0 = 1

2πi

∫ +∞

−∞trace − σ3

a∗

[ −δa bδa − aδb

aδb − bδa −δa

]∧ δS S−1 dλ.

After simple algebra we arrive at the stated identity. Theorem is proved. �

38 K. L. VANINSKY

Remark 1. The formula of theorem can be put easily into more familiar formusing the identities

|a|2 − |b|2 = 1, δ log |b|2 = 2|a|δ|a||a|2 − 1

.

Indeed,

1

i

δb(λ) ∧ δb(λ)

|a(λ)|2 = |b(λ)|2δ log |b(λ)|2 ∧ δph b(λ)

|a(λ)|2= 2δ log|a(λ)| ∧ δph b(λ).

Therefore,

ω0 = 1

πi

∫ +∞


|a(λ)|2 = 2

π

∫ +∞

−∞δ log|a(λ)| ∧ δph b(λ) dλ.

Remark 2. The formula

ωn = 1

πi

∫ +∞


|a(λ)|2 λn dλ, n = 1, 2, . . . ,

subject to the constrains Hk = const, k = 1, . . . , n, gives Darboux coordinates forhigher symplectic forms.

Acknowledgements

We conclude by expressing thanks to A. Its, H. McKean and I. Krichever for stim-ulating discussions. We are also greatful to anonymous referee for remarks thathelped to improve the presentation.

References

1. Arnold, V. I.: Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.2. Bikbaev, R. F. and Kuksin, S. B.: On the parametrization of finite-gap solutions by frequency

vector and wavenumber vector and a theorem of I. Krichever, Lett. Math. Phys. 166 (1994),115–122.

3. D’Hooker, E., Krichever, I. M. and Phong, D. H.: Seiberg–Witten theory, symplectic forms,and Hamiltonian theory of solitons, Preprint hep-th 0212313. Beijing and Hangzhou, 2002.

4. Faddeev, L. D. and Takhtadzian, L. A.: Hamiltonian Methods in the Theory of Solitons,Springer-Verlag, 1987.

5. Faddeev, L. D. and Zakharov, V. E.: Korteweg–de Vries equation: A completely integrableHamiltonian system, Functional Anal. Appl. 5 (1971), 18–27.

6. Flashka, H. and McLaughlin, D.: Canonically conjugate variables for the Korteweg–de Vriesequation and the Toda lattice with the periodic boundary conditions, Progr. Theoret. Phys. 55(2)(1976), 438–456.

7. Its, A. R.: The Liouville theorem and the inverse scattering method, J. Soviet. Math. 31(6)(1985), 299–334.


8. Krein, M. G.: On the theory of accelerant and S-matrices of canonical differential equations,Dokl. Akad. Nauk SSSR 111 (1956), 1167–1170.

9. Krein, M. G. and Melik-Adamian, P. E.: A contribution to the theory of S-matrices of canonicaldifferential equations with summable potential, Dokl. Akad. Nauk Armjan. SSR 46 (1968), 150–155.

10. Krichever, I. M.: Methods of algebraic geometry in the theory of nonlinear equations, UspekhiMat. Nauk 32(6) (1977), 183–208.

11. Krichever, I. M.: Elliptic solutions to the difference nonlinear equations and nested Bethe ansatzequations, In: Calogero–Moser–Sutherland models, (Montreal, QC, 1997)), CRM Ser. Math.Phys., Springer, New York, 2000.

12. Krichever, I. M.: Perturbation theory in periodic problems for two-dimensional integrablesystems, In: Soviet Sci. Rev. Sect. C. Math. Phys. 9, 1992, pp. 1–103.

13. Krichever, I. M. and Phong, D. H.: Symplectic Forms in the Theory of Solitons, Surveys inDifferential Geometry: Integrable Systems 4, International Press, Boston, 1999; Preprint hep-th9708170.

14. Kulish, P. P. and Reiman, A. G.: Hierarchy of symplectic forms for the Schrödinger and Diracequations on a line, Zap. LOMI 77 (1978), 134–147.

15. Kuznetzov, V. B., Nijhoff, F. W. and Sklyanin, E. K.: Separation of variables for the Ruijsenaarssystem, CMP 189(3) (1997), 855–877.

16. Melik-Adamian, P. E.: On the properties of S-matrices of canonical differential equations onthe entire line, Dokl. Akad. Nauk Armjan. SSR 58 (1974), 199–205.

17. McKean, H. P.: Trace formulas and the canonical 1-form, Algebraic Aspects of IntegrableSystems, In: Progr. Nonlinear Differential Equations 26, Birkhäuser, Boston, MA, 1997,pp. 217–235.

18. McKean, H. P.: A novel aspect of action-angle variables, In: Proc. Symp. Pure Math. 65, 1999,pp. 155–160.

19. McKean, H. P. and Vaninsky, K. L.: Action-angle variables for the cubic Schrödinger equation,Comm. Pure Appl. Math. L (1997), 489–562.

20. Moser, J.: Integrable Hamiltonian systems and spectral theory, In: Lezioni Fermiane, ScuolaNorm. Sup., Piza, 1983.

21. Novikov, S. P., Manakov, S. V. and Pitaevsky, L. P.: Theory of Solitons. The Inverse ScatteringMethod (Trans. from Russian). Consultants Bureau [Plenum], New York, 1984.

22. Novikov, S. P. and Veselov, A. P.: Poisson brackets and complex tori, Trudy Mat. Inst. Steklov.165 (1984), 49–61.

23. Seiberg, N. and Witten, E.: Electro-magnetic duality, monopole condensation, and confine-ment in N = 2 supersymmetric Yang–Mills theory, Nuclear Phys. B 426 (1994), 19–53,hep-th/9407087.

24. Seiberg, N. and Witten, E.: Monopoles, duality, and chiral symmetry breaking in N = 2supersymmetric QCD, Nuclear Phys. B 431 (1994), 494, hep-th/9410167.

25. Sklyanin, E. K.: Separation of variables – new trends, Quantum field theory, integrable modelsand beyond (Kyoto, 1994), Progr. Theoret. Phys. Suppl. 118 (1995), 35–60.

26. Vaninsky, K. L.: A convexity theorem in the scattering theory for the Dirac operator, Trans.Amer. Math. Soc. 350 (1998), 1895–1911.

27. Vaninsky, K. L.: The scattering divisor for the cubic Schrödinger equation (paper in prepara-tion).

28. Zakharov, V. E. and Manakov, S. V.: On the complete integrability of a nonlinear Schrödingerequation, Teoret. and Math. Phys. 19 (1974), 332–343.

Mathematical Physics, Analysis and Geometry (2005) 8: 41–58 © Springer 2005DOI: 10.1007/s11040-004-0936-z

The Singularity of Kontsevich’s Solution forQH∗(CP2)

DAVIDE GUZZETTIResearch Institute for Mathematical Sciences (RIMS), Kyoto University, Kitashirakawa, Sakyo-ku,Kyoto 606-8502, Japan. e-mail: [email protected]

(Received: 30 December 2002)

Abstract. In this paper we study the nature of the singularity of the Kontsevich’s solution of theWDVV equations of associativity. We prove that it corresponds to a singularity in the change of twocoordinates systems of the Frobenius manifold given by the quantum cohomology of CP2.

Mathematics Subject Classifications (2000): 53C99, 32D99, 14N35, 88A99.

Key words: WDVV equation, Frobenius manifold, quantum cohomology.

1. Introduction

In this paper we study the nature of the singularity of the solution of the WDVVequations of associativity for the quantum cohomology of the complex projectivespace CP2. As we will explain in detail below, the quantum cohomology of aprojective space CPd (d integer) is a Frobenius manifold which has a structurespecified by a solution to a WDVV equation. In the case of CP2 such a solution wasfound by Kontsevich [20] in the form of a convergent series in the flat coordinates(t1, t2, t3) of the corresponding Frobenius manifold:

F(t) := 12 [(t1)2t3 + t1(t2)2] + 1

t3

∞∑

k=1

Ak[(t3)3 exp(t2)]k, Ak ∈ R. (1)

The series converges in a neighborhood of (t3)3 exp(t2) = 0 with a certain radius ofconvergence estimated by Di Francesco and Itzykson [5]. The coefficients Ak arereal and are the Gromov–Witten invariants of genus zero. We will explain this pointlater. As for the Gromov–Witten invariants of genus one of CP2, we refer to [13],where B. Dubrovin and Y. Zhang proved that their G-function has the same radiusof convergence of (1).

As we will explain below, the nature of the boundary points of the ball ofconvergence of (1) is important to study of the global structure of the manifold.

In the following, we first state rigorously the problem of the global structureof a Frobenius manifold, then we introduce the quantum cohomology of CPd asa Frobenius manifold and we explain its importance in enumerative geometry.

42 DAVIDE GUZZETTI

Finally, we study the boundary points of the ball of convergence of Kontsevich’ssolution. We prove that they correspond to a singularity in the change of twocoordinates systems.

Our paper is part of a project to study of the global structure of Frobeniusmanifolds that we started in [15].

1.1. FROBENIUS MANIFOLDS AND THEIR GLOBAL STRUCTURE

The subject of this subsection can be found in [9, 10] or, in a more synthetic way,in [15].

The WDVV equations of associativity were introduced by Witten [28], Dijk-graaf, Verlinde E., Verlinde H. [6]. They are differential equations satisfied by theprimary free energy F(t) in two-dimensional topological field theory. F(t) is afunction of the coupling constants t := (t1, t2, . . . , tn) t i ∈ C. Let ∂α := ∂

∂tα.

Given a nondegenerate symmetric matrix ηαβ , α, β = 1, . . . , n, and numbersq1, q2, . . . , qn, r1, r2, . . . , rn, d, (rα = 0 if qα �= 1, α = 1, . . . , n), the WDVVequations are

∂α∂β∂λFηλµ∂µ∂γ ∂δF = the same with α, δ exchanged, (2)

∂1∂α∂βF = ηαβ, (3)

E(F) = (3 − d)F + (at most) quadratic terms, (4)

where the matrix (ηαβ) is the inverse of the matrix (ηαβ) and the differential operatorE is

E :=n∑

α=1

Eα∂α, Eα := (1 − qα)tα + rα, α = 1, . . . , n,

and will be called Euler vector field.Frobenius structures first appeared in the works of K. Saito [25, 26] with the

name of flat scructures. The complete theory of Frobenius manifolds was thendeveloped by B. Dubrovin as a geometrical setting for topological field theory andWDVV equations in [7]. Such a theory has links to many branches of mathematicslike singularity theory and reflection groups [25, 26, 12, 9], algebraic and enu-merative geometry [20, 22], isomonodromic deformations theory, boundary valueproblems and Painlevé equations [10].

If we define cαβγ (t) := ∂α∂β∂γ F (t), cγ

αβ(t) := ηγµcαβµ(t) (sum over repeatedindices is always omitted in the paper), and we consider a vector space A =span(e1, . . . , en), then we obtain a family of commutative algebras At with themultiplication eα · eβ := c

γ

αβ(t)eγ . Equation (2) is equivalent to associativity and(3) implies that e1 is the unity.

DEFINITION. A Frobenius manifold is a smooth/analytic manifold M over Cwhose tangent space TtM at any t ∈ M is an associative, commutative algebra

THE SINGULARITY OF KONTSEVICH’S SOLUTION 43

with unity e. Moreover, there exists a nondegenerate bilinear form 〈 , 〉 defining aflat metric (flat means that the curvature associated to the Levi–Civita connectionis zero).

We denote by · the product and by ∇ the covariant derivative of 〈·, ·〉. We requirethat the tensors c(u, v, w) := 〈u · v, w〉, and ∇yc(u, v, w), u, v, w, y ∈ TtM , besymmetric. Let t1, . . . , tn be (local) flat coordinates for t ∈ M . Let eα := ∂α

be the canonical basis in TtM , ηαβ := 〈∂α, ∂β〉, cαβγ (t) := 〈∂α · ∂β, ∂γ 〉. Thesymmetry of c corresponds to the complete symmetry of ∂δcαβγ (t) in the indices.This implies the existence of a function F(t) such that ∂α∂β∂γ F (t) = cαβγ (t)

satisfying the WDVV (2). Equation (3) follows from the axiom ∇e = 0 whichyields e = ∂1. Some more axioms are needed to formulate the quasi-homogeneitycondition (4) and we refer the reader to [9–11]. In this way the WDVV equationsare reformulated in a geometrical terms.

We first consider the problem of the local structure of Frobenius manifolds.A Frobenius manifold is characterized by a family of flat connections ∇(z) para-meterized by a complex number z, such that for z = 0 the connection is associatedto 〈 , 〉. For this reason ∇(z) are called deformed connections. Let u, v ∈ TtM ,ddz

∈ TzC; the family is defined on M × C as follows:

∇uv := ∇uv + zu · v,

∇ ddz

v := ∂

∂zv + E · v − 1

zµv,

∇ ddz

d

dz= 0, ∇u

d

dz= 0,

where E is the Euler vector field and

µ := I − d

2− ∇E

is an operator acting on v. In flat coordinates t = (t1, . . . , tn), µ becomes:

µ = diag(µ1, . . . , µn), µα = qα − d

2,

provided that ∇E is diagonalizable. This will be assumed in the paper. A flatcoordinate t (t, z) is a solution of ∇dt = 0, which is a linear system

∂αξ = zCα(t)ξ, (5)

∂zξ =[U(t) + µ

z

]ξ, (6)

where ξ is a column vector of components ξα = ηαµ ∂t∂tµ

, α = 1, . . . , n andCα(t) := (cβ

αγ (t)), U := (Eµcβµγ (t)) are n × n matrices.

44 DAVIDE GUZZETTI

The quantum cohomology of projective spaces, to be introduce below, belongsto the class of semi-simple Frobenius manifolds, namely analytic Frobenius man-ifolds such that the matrix U can be diagonalized with distinct eigenvalues on anopen dense subset M of M . Then, there exists an invertible matrix φ0 = φ0(t) suchthat φ0Uφ−1

0 = diag(u1, . . . , un) =: U , ui �= uj for i �= j on M. The systems (5)and (6) become:

∂y

∂ui

= [zEi + Vi]y, (7)

∂y

∂z=

[U + V

z

]y, (8)

where the row-vector y is y := φ0ξ , Ei is a diagonal matrix such that (Ei)ii = 1and (Ei)jk = 0 otherwise, and

Vi := ∂φ0

∂ui

φ−10 , V := φ0µφ−1

0 .

As it is proved in [9, 10], u1, . . . , un are local coordinates on M. The two bases∂

∂tν, ν = 1, . . . , n and ∂

∂ui, i = 1, . . . , n are related by φ0 according to the linear

combination ∂∂tν

= ∑ni=1

(φ0)iν(φ0)i1

∂∂ui

. Locally we obtain a change of coordinates, tα =tα(u), then φ0 = φ0(u), V = V (u). The local Frobenius structure of M is given byparametric formulae:

tα = tα(u), F = F(u), (9)

where tα(u), F(u) are certain meromorphic functions of (u1, . . . , un), ui �= uj ,which can be obtained from φ0(u) and V (u). Their explicit construction was theobject of [15]. We stress here that the condition ui �= uj is crucial. We will furthercomment on this when we face the problem of the global structure.

The dependence of the system on u is isomonodromic. This means that themonodromy data of the system (8), to be introduced below, do not change fora small deformation of u. Therefore, the coefficients of the system in every localchart of M are naturally labeled by the monodromy data. To calculate the functions(9) in every local chart one has to reconstruct the system (8) from its monodromydata. This is the inverse problem.

We briefly explain what are the monodromy data of the system (8) and why theydo not depend on u (locally). For details the reader is referred to [10]. At z = 0the system (8) has a fundamental matrix solution (i.e. an invertible n × n matrixsolution) of the form

Y0(z, u) =[ ∞∑

p=0

φp(u)zp

]zµzR, (10)


where Rαβ = 0 if µα − µβ �= k > 0, k ∈ N. At z = ∞ there is a formal n × n

matrix solution of (8) given by

YF =[I + F1(u)

z+ F2(u)

z2+ · · ·

]ezU ,

where Fj(u)’s are n × n matrices. It is a well known result that there exist fun-damental matrix solutions with asymptotic expansion YF as z → ∞ [2]. Let l bea generic oriented line passing through the origin. Let l+ be the positive half-lineand l− the negative one. Let �L and �R be two sectors in the complex plane tothe left and to the right of l respectively. There exist unique fundamental matrixsolutions YL and YR having the asymptotic expansion YF for x → ∞ in �L and�R, respectively [2]. They are related by an invertible connection matrix S, calledStokes matrix, such that YL(z) = YR(z)S for z ∈ l+. As it is proved in [10] we alsohave YL(z) = YR(z)ST on l−.

Finally, there exists a n × n invertible connection matrix C such that Y0 = YRC

on �R.

DEFINITION. The matrices R, C, µ and the Stokes matrix S of the system (8)are the monodromy data of the Frobenius manifold in a neighborhood of the pointu = (u1, . . . , un). It is also necessary to specify which is the first eigenvalue of µ,because the dimension of the manifold is d = −2µ1 (a more precise definition ofmonodromy data is in [10]).

The definition makes sense because the data do not change if u undergoes asmall deformation. This problem is discussed in [10]. We also refer the readerto [17] for a general discussion of isomonodromic deformations. Here we justobserve that since a fundamental matrix solution Y (z, u) of (8) also satisfies (7),then the monodromy data can not depend on u (locally). In fact, ∂Y

∂uiY−1 = zEi +Vi

is single-valued in z.The inverse problem can be formulated as a boundary value problem (b.v.p.).

Let’s fix u = u(0) = (u(0)

1 , . . . , u(0)n ) such that u

(0)i �= u

(0)j for i �= j . Suppose

we give µ, µ1, R, an admissible line l, S and C. Some more technical conditionsmust be added, but we refer to [10]. Let D be a disk specified by |z| < ρ for somesmall ρ. Let PL and PR be the intersection of the complement of the disk with�L and �R, respectively. We denote by ∂DR and ∂DL the lines on the boundaryof D on the side of PR and PL, respectively; we denote by l+ and l− the portionof l+ and l− on the common boundary of PR and PL. Let’s consider the followingdiscontinuous b.v.p.: we want to construct a piecewise holomorphic n × n matrixfunction

(z) ={

R(z), z ∈ PR, L(z), z ∈ PL, 0(z), z ∈ D

continuous on the boundary PR, PL, D respectively, such that

L(ζ ) = R(ζ )eζUSe−ζU , ζ ∈ l+,

46 DAVIDE GUZZETTI

L(ζ ) = R(ζ )eζUST e−ζU , ζ ∈ l−,

0(ζ ) = R(ζ )eζUCζ−Rζ−µ, ζ ∈ ∂DR,

0(ζ ) = L(ζ )eζUS−1Cζ−Rζ−µ, ζ ∈ ∂DL,

L/R(z) → I if z → ∞ in PL/R.

The reader may observe that YL/R(z) := L/R(z)ezU , Y (0)(z) := 0(z, u)zµzR

have precisely the monodromy properties of the solutions of (8).

THEOREM [23, 21, 10]. If the above boundary value problem has solution for agiven u(0) = (u

(0)1 , . . . , u(0)

n ) such that u(0)i �= u

(0)j for i �= j , then:

(i) It is unique.(ii) The solution exists and it is analytic for u in a neighborhood of u(0).

(iii) The solution has analytic continuation as a meromorphic function on the uni-versal covering of Cn\{diagonals}, where “diagonals” stands for the union ofall the sets {u ∈ Cn | ui = uj , i �= j }.

A solution YL/R, Y (0) of the b.v.p. solves the system (7), (8). This means thatwe can locally reconstruct V (u), φ0(u) and (9) from the local solution of the b.v.p.It follows that every local chart of the atlas covering the manifold is labeled bymonodromy data. Moreover, V (u), φ0(u) and (9) can be continued analytically asmeromorphic functions on the universal covering of Cn\diagonals.

Let Sn be the symmetric group of n elements. Local coordinates (u1, . . . , un)

are defined up to permutation. Thus, the analytic continuation of the local struc-ture of M is described by the braid group Bn, namely the fundamental groupof (Cn\diagonals)/Sn. There exists an action of the braid group itself on themonodromy data, corresponding to the change of coordinate chart. The group isgenerated by n−1 elements β1, . . . , βn−1 such that βi is represented as a deforma-tion consisting of a permutation of ui , ui+1 moving counter-clockwise (clockwiseor counter-clockwise is a matter of convention).

If u1, . . . , un are in lexicographical order w.r.t. l, so that S is upper triangular,the braid βi acts on S as follows [10]:

S → Sβi = Ai(S)SAi(S),

where

(Ai(S))kk = 1, k = 1, . . . , n, n �= i, i + 1,

(Ai(S))i+1,i+1 = −si,i+i ,

(Ai(S))i,i+1 = (Ai(S))i+1,i = 1


and all the other entries are zeros. For a generic braid β the action S → Sβ isdecomposed into a sequence of elementary transformations as above. In this way,we are able to describe the analytic continuation of the local structure in terms ofmonodromy data.

Not all the braids are actually to be considered. Suppose we do the followinggauge y → Jy, J = diag(±1, . . . , ±1), on the system (8). Therefore JUJ −1 ≡ U

but S is transformed to JSJ−1, where some entries change sign. The formulaewhich define a local chart of the manifold in terms of monodromy data, which aregiven in [10, 15], are not affected by this transformation. The analytic continuationof the local structure on the universal covering of (Cn\diagonals)/Sn is thereforedescribed by the elements of the quotient group

Bn/{β ∈ Bn | Sβ = JSJ }. (11)

From these considerations it is proved in [10] that:

THEOREM [10]. Given monodromy data (µ1, µ, R, S, C), the local Frobeniusstructure obtained from the solution of the b.v.p. extends to an open dense subsetof the covering of (Cn\diagonals)/Sn w.r.t. the covering transformations (11).

Let’s start from a Frobenius manifold M of dimension d. Let M be the opensub-manifold where U(t) has distinct eigenvalues. If we compute its monodromydata (µ1 = − d

2 , µ, R, S, C) at a point u(0) ∈ M and we construct the Frobeniusstructure from the analytic continuation of the corresponding b.v.p. on the cover-ing of (Cn\diagonals)/Sn w.r.t. the quotient (11), then there is an equivalence ofFrobenius structures between this last manifold and M.

To understand the global structure of a Frobenius manifold we have to study(9) when two or more distinct coordinates ui , uj , etc., merge. φ0(u), V (u) and (9)are multi-valued meromorphic functions of u = (u1, . . . , un) and the branchingoccurs when u goes around a loop around the set of diagonals

⋃ij {u ∈ Cn | ui =

uj , i �= j }. φ0(u), V (u) and (9) have singular behavior if ui → uj (i �= j ). Wecall such behavior critical behavior.

The Kontsevich’s solution introduced at the beginning has a radius of conver-gence which might be due to the fact that some coordinates ui , uj merge at theboundary of the ball of convergence. We will prove that this is not the case. Rather,there is a singularity in the change of coordinates u → t .

1.2. INTERSECTION FORM OF A FROBENIUS MANIFOLD

The deformed flat connection was introduced as a natural structure on a Frobeniusmanifold and allows to transform the problem of solving the WDVV equations toa problem of isomonodromic deformations. There is a further natural structure ona Frobenius manifold which makes it possible to do the same. It is the intersectionform. We need it as a tool to calculate the canonical coordinates later.

48 DAVIDE GUZZETTI

There is a natural isomorphism ϕ: TtM → T ∗t M induced by 〈. , .〉. Namely, let

v ∈ TtM and define ϕ(v) := 〈v, .〉. This allows us to define the product in T ∗t M

as follows: for v, w ∈ TtM we define ϕ(v) · ϕ(w) := 〈v · w, .〉. In flat coordinatest1, . . . , tn the product is

dtα · dtβ = cαβγ (t)dtγ , cαβ

γ (t) = ηβδcαδγ (t)

(sums over repeated indices are omitted).

DEFINITION. The intersection form at t ∈ M is a bilinear form on T ∗t M defined

by

(ω1, ω2) := (ω1 · ω2)(E(t)),

where E(t) is the Euler vector field. In coordinates

gαβ(t) := (dtα, dtβ) = Eγ (t)cαβγ (t).

In the semi-simple case, let u1, . . . , un be local canonical coordinates, equal tothe distinct eigenvalues of U(t). From the definitions we have

dui · duj = 1

ηii

δij dui, gij (u) = (dui, duj ) = ui

ηii

δij , ηii = (φ0)2i1.

Then gij − ληij = ui−λ

ηiiδij and

det((gij − ληij )) = 1

det((ηij ))(u1 − λ)(u2 − λ) · · · (un − λ).

Namely, the roots λ of the above polynomial are the canonical coordinates.In order to compute gαβ , in the paper we are going to use the following formula.

We differentiate twice the expression

Eγ ∂γ F = (2 − d)F + 12Aαβtαtβ + Bαt

α + C

which is the quasi-homogeneity of F up to quadratic terms. By recalling that Eγ =(1 − qγ )tγ + rγ and that ∂α∂β∂γ F = cαβγ we obtain

gαβ(t) = (1 + d − qα − qβ)∂α∂βF (t) + Aαβ, (12)

where ∂α = ηαβ∂β , Aαβ = ηαγ ηβδAγδ.


2. Quantum Cohomology of Projective Spaces

In this section we introduce the Frobenius manifold called quantum cohomology ofthe projective space CPd and we describe its connections to enumerative geometry.

It is possible to introduce a structure of Frobenius algebra on the cohomologyH ∗(X, C) of a closed oriented manifold X of dimension d such that

Hi(X, C) = 0 for i odd.

Then

H ∗(X, C) =d⊕

i=0

H 2i(X, C).

For brevity we omit C in H . H ∗(X) can be realized by classes of closed differentialforms. The unit element is a 0-form e1 ∈ H 0(X). Let us denote by ωα a form inH 2qα (X), where q1 = 0, q2 = 1, . . . , qd+1 = d. The product of two forms ωα, ωβ

defined by the wedge product ωα ∧ ωβ ∈ H 2(qα+qβ)(X) and the bilinear form is

〈ωα, ωβ〉 :=∫

X

ωα ∧ ωβ �= 0 ⇐⇒ qα + qβ = d.

It is not degenerate by Poincaré duality and qα + qd−α+1 = d.Let X = CP d . Let e1 = 1 ∈ H 0(CP d), e2 ∈ H 2(CP d), . . . , ed+1 ∈ H 2d(CP d)

be a basis in H ∗(CP d). For a suitable normalization we have

(ηαβ) := (〈eα, eβ〉) =

11

. ..

11

.

The multiplication is

eα ∧ eβ = eα+β−1.

We observe that it can also be written as

eα ∧ eβ = cγ

αβeγ , sums on γ ,

where

ηαδcδβγ := ∂3F0(t)

∂tα∂tβ∂tγ,

F0(t) := 12(t

1)2tn + 1

2t1

n−1∑

α=2

tαtn−α+1.

50 DAVIDE GUZZETTI

F0 is the trivial solution of WDVV equations. We can construct a trivial Frobeniusmanifold whose points are t := ∑d+1

α=1 tαeα. It has tangent space H ∗(CPd) at any t .By quantum cohomology of CPd (denoted by QH∗(CPd)) we mean a Frobeniusmanifold whose structure is specified by

F(t) = F0(t) + analytic perturbation.

This manifold has therefore tangent spaces TtQH∗(CPd) = H ∗(CPd), with thesame 〈. , .〉 as above, but the multiplication is a deformation, depending on t , of thewedge product (this is the origin of the adjective “quantum”).

3. The Case of CP2

To start with, we restrict to CP2. In this case

F0(t) = 12 [(t1)2t3 + t1(t2)2]

which generates the product for the basis e1 = 1 ∈ H 0, e2 ∈ H 2, e3 ∈ H 4. Thedeformation was introduced by Kontsevich [20].

3.1. KONTSEVICH’S SOLUTION

The WDVV equations for n = 3 variables have solutions

F(t1, t2, t3) = F0(t1, t2, t3) + f (t2, t3).

f (t2, t3) satisfies a differential equation obtained by substituting F(t) into theWDVV equations. Namely:

f222f233 + f333 = (f223)2 (13)

with the notation fijk := ∂3f

∂ti∂tj ∂tk. As for notations, the variables tj are flat coordi-

nates in the Frobenius manifold associate to F . They should be written with upperindices, but we use the lower for convenience of notation.

Let Nk be the number of rational curves CP1 → CP2 of degree k through 3k−1generic points. Kontsevich [20] constructed the solution

f (t2, t3) = 1

t3ϕ(τ), ϕ(τ) =

∞∑

k=1

Akτk, τ = t3

3 et2, (14)

where

Ak = Nk

(3k − 1)! .


The Ak (or Nk) are called Gromov–Witten invariants of genus zero. We note thatthis solution has precisely the form of the general solution of the WDVV equationsfor n = 3, d = 2 and r2 = 3 [9]. If we put τ = eX and we define

(X) := ϕ(eX) =∞∑

k=1

AkekX

we rewrite (13) as follows:

−6 + 33 ′ − 54 ′′ − ( ′′)2 + ′′′(27 + 2 ′ − 3 ′′) = 0. (15)

The prime stands for the derivative w.r.t. X. If we fix A1, the above (15) determinesthe Ak uniquely. Since N1 = 1, we fix

A1 = 12 .

Then (15) yields the recurrence relation

Ak =k−1∑

i=1

[AiAk−1i(k − i)((3i − 2)(3k − 3i − 2)(k + 2) + 8k − 8)

6(3k − 1)(3k − 2)(3k − 3)

]. (16)

The convergence of (14) was studied by Di Francesco and Itzykson [5]. Theyproved that

Ak = bakk− 72

(1 + O

(1

k

)), k → ∞

and numerically estimated

a = 0.138, b = 6.1.

We remark that the problem of the exact computation of a and b is open. Theresult implies that ϕ(τ) converges in a neighborhood of τ = 0 with radius ofconvergence 1

a.

We remark that as far as the Gromov–Witten invariants of genus one are con-cerned, B. Dubrovin and Y. Zhang proved in [13] that their G-function has thesame radius of convergence of (1). Moreover, they proved the asymptotic formulafor such invariants as conjectuder by Di Francesco and Itzykson. As far as I know,such a result was explained in lectures, but not published.

The proof of [5] is divided in two steps. The first is based on the relation (16),to prove that

A1k

k → a for k → ∞, 1108 < a < 2

3 .

a is real positive because the Ak’s are such. It follows that we can rewrite

Ak = bakkω

(1 + O

(1

k

)), ω ∈ R.

52 DAVIDE GUZZETTI

The above estimate implies that ϕ(τ) has the radius of convergence 1a. The second

step is the determination of ω making use of the differential equation (15). Let’swrite

Ak := Ckak,

(X) =∞∑

k=1

AkekX =

∞∑

k=1

Ckek(X−X0), X0 := ln

1

a.

The inequality 1108 < a < 2

3 implies that X0 > 0. The series converges at least for�X < X0. To determine ω we divide (X) into a regular part at X0 and a singularone. Namely

(X) =∞∑

k=0

dk(X − X0)k + (X − X0)

γ

∞∑

k=0

ek(X − X0)k, γ > 0, γ /∈ N,

dk and ek are coefficients. By substituting into (15) we see that the equation issatisfied only if γ = 5

2 . Namely:

(X) = d0 + d1(X − X0) + d2(X − X0)2 + e0(X − X0)

52 + · · · .

This implies that (X), ′(X) and ′′(X) exist at X0 but ′′′(X) diverges like

′′′(X) � 1√X − X0

, X → X0. (17)

On the other hand ′′′(X) behaves like the series∞∑

k=1

bkω+3ek(X−X0), �(X − X0) < 0.

Suppose X ∈ R, X < X0. Let us put � := X − X0 < 0. The above series is

b

|�|3+ω

∞∑

k=1

(|�|k)3+ωe−|�|k ∼ b

|�|3+ω

∫ ∞

0dx x3+ωe−x.

It follows from (17) that this must diverge like �− 12 , and thus ω = − 7

2 (the integralremains finite).

As a consequence of (15) and of the divergence of ′′′(X)

27 + 2 ′(X0) − 3 ′′(X0) = 0.


4. The Case of CPd

The case d = 1 is trivial, the deformation being:

F(t) = 12 t

21 t2 + et2 .

For any d � 2, the deformation is given by the following solution of the WDVVequations [20, 22]:

F(t) = F0(t) +∞∑

k=1

[ ∞∑

n=2

∼∑

α1,...,αn

Nk(α1, . . . , αn)

n! tα1,...,αn

]ekt2,

where∼∑

α1,...,αn

:=∑

α1+···+αn=2n+d(k+1)+k−3

.

Here Nk(α1, . . . , αn) is the number of rational curves CP1 → CPd of degree k

through n projective subspaces of codimensions α1 − 1, . . . , αn − 1 � 2 in generalposition. In particular, there is one line through two points, then

N1(d + 1, d + 1) = 1.

Note that in Kontsevich solution Nk = Nk(d + 1, d + 1).In flat coordinates the Euler vector field is

E =∑

α �=2

(1 − qα)tα ∂

∂tα+ k

∂

∂t2,

q1 = 0, q2 = 1, q3 = 2, . . . , qk = k − 1

and

µ = diag(µ1, . . . , µk)

= diag

(−d

2, −d − 2

2, . . . ,

d − 2

2,d

2

), µα = qα − d

2.

5. Nature of the Singular Point X0 = ln(1/a)

We are now ready to formulate the problem of the paper. We need to investigatethe nature of the singularity X0, namely whether it corresponds to the fact thattwo canonical coordinates u1, u2, u3 merge. Actually, we pointed out that thestructure of the semi-simple manifold may become singular in such points becausethe solutions of the boundary value problem are meromorphic on the universalcovering of Cn\diagonals and are multi valued if ui − uj (i �= j ) goes around a

54 DAVIDE GUZZETTI

loop around zero. We will verify that actually ui , uj do not merge, but the changeof coordinates u → t is singular at X0. In this section we restore the upper indicesfor the flat coordinates tα.

The canonical coordinates can be computed from the intersection form. Werecall that the flat metric is

η = (ηαβ) :=( 0 0 1

0 1 01 0 0

).

The intersection form is given by the formula (12):

gαβ = (d + 1 − qα − qβ)ηαµηβν∂µ∂νF + Aαβ, α, β = 1, 2, 3,

where d = 2 and the charges are q1 = 0, q2 = 1, q3 = 2. The matrix Aαβ appearsin the action of the Euler vector field

E := t1∂1 + 3∂2 − t3∂3

on F(t1, t2, t3):

E(F)(t1, t2, t3) = (3 − d)F (t1, t2, t3) + Aµνtµtν ≡ F(t1, t2, t3) + 3t1t2.

Thus

(Aαβ) = (ηαµηβνAµν) =( 0 0 0

0 0 30 3 0

).

After the above preliminaries, we are able to compute the intersection form:

(gαβ) =

3

[t3]3 [2 − 9 ′ + 9 ′′] 2[t3]2 [3 ′′ − ′] t1

2[t3]2 [3 ′′ − ′] t1 + 1

t3 ′′ 3

t1 3 −t3

.

The canonical coordinates are roots of

det((gαβ − uη) = 0.

This is the polynomial

u3 −(

3t1 + 1

t3 ′′

)u2 −

−(

−3[t1]2 − 2t1

t3 ′′ + 1

[t3]2(9 ′′ + 15 ′ − 6 )

)u + P(t, ),

where

P(t, ) = 1

[t3]3(−9t1t3 ′′ + 243 ′′ − 243 ′ + 6 ′ − 9( ′′)2 +

+ 6t1t3 + [t1]2[t3]2 ′′ − 3 ′ ′′ + [t1]3[t3]3 −

− 4( ′)2 + 54 − 15t1t3 ′).


It follows that

ui(t1, t3, X) = t1 + 1

t3Vi (X).

Vi(X) depends on X through (X) and derivatives. We also observe that

u1 + u2 + u3 = 3t1 + 1

t3 ′′(X).

As a first step, we verify numerically that ui �= uj for i �= j at X = X0.In order to do this we need to compute (X0), ′(X0), ′′(X0) in the followingapproximation

(X0) ∼=N∑

k=1

Ak

1

ak, ′(X0) ∼=

N∑

k=1

kAk

1

ak, ′′(X0) ∼=

N∑

k=1

k2Ak

1

ak.

We fixed N = 1000 and we computed the Ak, k = 1, 2, . . . , 1000 exactly usingthe relation (16). Then we computed a and b by the least squares method. For largek, say for k � N0, we assumed that

Ak∼= bakk− 7

2 (18)

which implies

ln(Akk72 ) ∼= (ln a)k + ln b.

The corrections to this law are O( 1k). This is the line to fit the data k

72 Ak. Let

y := 1

N − N0 + 1

N∑

k=N0

ln(Akk72 ), k := 1

N − N0 + 1

N∑

N0

k.

By the least squares method

ln a =∑N

k=N0(k − k)(ln(Akk

72 ) − y)

∑Nk=N0

(k − k)2, with error

(1

k2

),

ln b = y − (ln a)k, with error

(1

k

).

For N = 1000, A1000 is of the order 10−840. In our computation we set the accuracyto 890 digits. Here is the result, for three choices of N0. The result should improveas N0 increases, since the approximation (18) becomes better.

N0 = 500, a = 0.138009444 . . . , b = 6.02651 . . .

N0 = 700, a = 0.138009418 . . . , b = 6.03047 . . .

N0 = 900, a = 0.138009415 . . . , b = 6.03062 . . .

56 DAVIDE GUZZETTI

It follows that (for N0 = 900)

(X0) = 4.268908 . . . , ′(X0) = 5.408 . . . , ′′(X0) = 12.25 . . .

With these values we find

27 + 2 ′(X0) − 3 ′′(X0) = 1.07 . . . ,

but the above should vanish! The reason why this does not happen is that ′′(X0) =∑Nk=1 k2Ak

1ak converges slowly. To obtain a better approximation we compute it

numerically as

′′(X0) = 13(27 + 2 ′(X0)) = 1

3

(27 + 2

N∑

k=1

kAk

1

ak

)= 12.60 . . .

Substituting into gαβ and setting t1 = t3 = 1 we find

u1 ≈ 22.25 . . . , u2 ≈ −(3.5 . . .) − (2.29 . . .)i, u3 = u2,

where i = √−1 and the bar means complex conjugation. Thus, with a sufficientaccuracy, we have verified that ui �= uj for i �= j .

We now prove that the singularity is a singularity for the change of coordinates

(u1, u2, u3) −→ (t1, t2, t3).

We recall that

∂u1

∂tα= (φ0)iα

(φ0)i1.

This may become infinite if (φ0)i1 = 0 for some i. In our case

u1 + u2 + u3 = 3t1 + 1

t3 (X)′′,

∂X

∂t1= 0,

∂X

∂t2= 1,

∂X

∂t3= 3

t3

and

∂

∂t1(u1 + u2 + u3) = 3,

∂

∂t2(u1 + u2 + u3) = 1

t3 (X)′′′,

∂

∂t3(u1 + u2 + u3) = 1

[t3]2 (X)′′ + 3

[t3]2 (X)′′′.

The above proves that the change of coordinates is singular because both ∂

∂t2 (u1 +u2 + u3) and ∂

∂t3 (u1 + u2 + u3) behave like (X)′′′ � 1√X−X0

for X → X0.


Acknowledgments

I thank A. Its and P. Bleher for suggesting me to try the computations of thispaper and for discussions. I thank B. Dubrovin for introducing me to the theoryof Frobenius manifolds and for discussing together the problem of this paper.

References

1. Anosov, D. V. and Bolibruch, A. A.: The Riemann–Hilbert Problem, Publication from theSteklov Institute of Mathematics, 1994.

2. Balser, W., Jurkat, W. B. and Lutz, D. A.: Birkhoff invariants and Stokes’ multipliers formeromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), 48–94.

3. Balser, W., Jurkat, W. B. and Lutz, D. A.: On the reduction of connection problems for differ-ential equations with an irregular singular point to ones with only regular singularities, SIAMJ. Math. Anal. 12 (1981), 691–721.

4. Birman, J. S.: Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, PrincetonUniv. Press, 1975.

5. Di Francesco, P. and Itzykson, C.: Quantum intersection rings, In: R. Dijkgraaf, C. Faber andG. B. M. van der Geer (eds), The Moduli Space of Curves, 1995.

6. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Nuclear Phys. B 352 (1991), 59.7. Dubrovin, B.: Integrable systems in topological field theory, Nuclear Phys. B 379 (1992), 627–

689.8. Dubrovin, B.: Geometry and itegrability of topological-antitopological fusion, Comm. Math.

Phys. 152 (1993), 539–564.9. Dubrovin, B.: Geometry of 2D topological field theories, In: Lecture Notes in Math. 1620,

1996, pp. 120–348.10. Dubrovin, B.: Painlevé trascendents in two-dimensional topological field theory, In: R. Conte

(ed.), The Painlevé Property, One Century Later, Springer, 1999.11. Dubrovin, B.: Geometry and Analytic Theory of Frobenius Manifolds, math.AG/9807034,

(1998).12. Dubrovin, B.: Differential geometry on the space of orbits of a Coxeter group,

math.AG/9807034 (1998).13. Dubrovin, B. and Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-

loop approximation, Comm. Math. Phys. 198 (1998), 311–361.14. Guzzetti, D.: Stokes matrices and monodromy for the quantum cohomology of projective

spaces, Comm. Math. Phys. 207 (1999), 341–383. Also see preprint math/9904099.15. Guzzetti, D.: Inverse problem and monodromy data for 3-dimensional Frobenius manifolds,

J. Math. Phys., Analysis and Geometry 4 (2001), 254–291.16. Its, A. R. and Novokshenov, V. Y.: The isomonodromic deformation method in the theory of

Painlevé equations, In: Lecture Notes in Math. 1191, 1986.17. Jimbo, M., Miwa, T. and Ueno, K.: Monodromy preserving deformations of linear ordinary

differential equations with rational coefficients (I), Phys. D 2 (1981), 306.18. Jimbo, M. and Miwa, T.: Monodromy preserving deformations of linear ordinary differential

equations with rational coefficients (II), Phys. D 2 (1981), 407–448.19. Jimbo, M. and Miwa, T.: Monodromy preserving deformations of linear ordinary differential

equations with rational coefficients (III), Phys. D 4 (1981), 26.20. Kontsevich, M. and Manin, Y. I.: Gromov–Witten classes, quantum cohomology and enumera-

tive geometry, Comm. Math. Phys. 164 (1994), 525–562.21. Malgrange, B.: Équations différentielles à coefficientes polynomiaux, Birkhäuser, 1991.

58 DAVIDE GUZZETTI

22. Manin, V. I.: Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, Max PlanckInstitut für Mathematik, Bonn, Germany, 1998.

23. Miwa., T: Painlevé property of monodromy preserving equations and the analyticity ofτ -functions, Publ. Res. Inst. Math. Sci. 17 (1981), 703–721.

24. Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology, Math. Res. Lett. 1(1994), 269–278.

25. Saito, K.: Preprint RIMS-288 (1979) and Publ. Res. Inst. Math. Sci. 19 (1983), 1231–1264.26. Saito, K., Yano, T. and Sekeguchi, J.: Comm. Algebra 8(4) (1980), 373–408.27. Sato, M., Miwa, T. and Jimbo, M.: Holonomic quantum fields. II – The Riemann–Hilbert

problem, Publ. Res. Inst. Math. Sci. 15 (1979), 201–278.28. Witten, E.: Nuclear Phys. B 340 (1990), 281–332.


Boundary Value Problems for Boussinesq TypeSystems

A. S. FOKAS1 and B. PELLONI2

1Department of Applied Mathematics and Theoretical Physics, Cambridge University,Cambridge CB3 0WA, UK. e-mail: [email protected] of Mathematics, University of Reading, Reading RG6 6AX, UK.e-mail: [email protected]

(Received: 25 July 2003; in final form: 13 May 2004)

Abstract. We characterise the boundary conditions that yield a linearly well posed problem for theso-called KdV–KdV system and for the classical Boussinesq system. Each of them is a system oftwo evolution PDEs modelling two-way propagation of water waves. We study these problems withthe spatial variable in either the half-line or in a finite interval. The results are obtained by extendinga spectral transform approach, recently developed for the analysis of scalar evolution PDEs, to thecase of systems of PDEs.

The knowledge of the boundary conditions that should be imposed in order for the problem to belinearly well posed can be used to obtain an integral representation of the solution. This knowledgeis also necessary in order to conduct numerical simulations for the fully nonlinear systems.

Mathematics Subject Classifications (2000): 34A30, 34A34, 35F10.

Key words: water waves, PDE systems, boundary value problems, integral representations.

1. Introduction

We introduce a general approach for studying boundary value problems for systemsof evolution equations in one space variable. We place the emphasis on the analysisof linear systems, but we also indicate how this analysis, combined with classicalPDE techniques and with numerical investigations, can provide useful informationabout nonlinear systems. The general methodology will be illustrated for the so-called KdV–KdV system [3]

ηt + ux + (uη)x + uxxx = 0,

ut + ηx + uux + ηxxx = 0,(1.1)

and for the ‘classical’ Boussinesq system [19]

ηt + ux + (uη)x = 0,

ut + ηx + uux − uxxt = 0.(1.2)

60 A. S. FOKAS AND B. PELLONI

In this paper, we consider mainly the versions of these systems obtained bylinearising around η = 0, u = 0. We assume that sufficiently regular initialconditions, denoted by η0(x) and by u0(x), are prescribed at t = 0:

η(x, 0) = η0(x), u(x, 0) = u0(x).

We study these systems for x either on the half-line or on a finite interval. In theformer case, we assume the η0(x) and u0(x) decay sufficiently rapidly as x → ∞.

The linearised KdV–KdV system is

ηt + ux + uxxx = 0,

ut + ηx + ηxxx = 0,(1.3)

and the linearised classical Boussinesq system is

ηt + ux = 0, (1.4a)

ut + ηx − uxxt = 0. (1.4b)

Our main interest in studying such linear problems is the issue of well posedness.�

This issue has recently attracted considerable attention. Indeed, even if the aim is tostudy systems such as (1.1)–(1.4) numerically, one is immediately confronted withthe problem of how to identify which boundary conditions must be prescribed in or-der for the exact problem – as well as for the numerical problem – to be well posed.This difficulty is reflected in the fact that most of the existing results apply eitherto the Cauchy problem or to the problem with space-periodic boundary conditions.For general boundary value problems it is difficult to answer this question even inthe simple case that the linearised system decouples into two scalar equations (thisis indeed the case for Equations (1.3)).

It will be shown here that it is possible to overcome this difficulty by extendingto systems of PDEs the method recently introduced in the literature for scalarPDEs (see, for example, the review [7]). The strength of the method presentedis that: (a) it provides an algorithm for characterising how many and exactly whichboundary conditions one needs to prescribe for the problem to be well posed; (b) ityields an integral representation of the solution with explicit x- and t-dependence,involving an x-transform of the initial conditions and an appropriate t-transform ofthe given boundary conditions.

We consider boundary value problems posed on either the half-line x ∈ [0, ∞)

or a finite interval x ∈ [0, L], L a positive constant. We establish the followingwell-posedness results:

• KdV–KdV on [0, ∞): three boundary conditions must be prescribed at x = 0.These can be conditions for either η(x, t), or u(x, t), or their first or secondx-derivatives.

• KdV–KdV on [0, L]: three boundary conditions must be prescribed at x = 0and three boundary conditions must be prescribed at x = L. These can beconditions for either η(x, t), or u(x, t), or their first or second x-derivatives.

� We call a boundary value problem well posed if it admits a unique (smooth) solution.

BOUNDARY VALUE PROBLEMS FOR BOUSSINESQ TYPE SYSTEMS 61

• Classical Boussinesq on [0, ∞): one boundary condition must be prescribedat x = 0. This condition can be given either for η(x, t) or for any of thex-derivatives of u(x, t).

• Classical Boussinesq on [0, L]: one boundary condition at each end x = 0and x = L must be prescribed. This condition can be given either for η(x, t)

or for any of the x-derivatives of u(x, t).

The solution representation we obtain, in each case, can be expressed conciselyin the form

q(x, t) = 1

2π

∫�

eikxI−ω(k)tρ(k) dk, (1.5)

where q(x, t) is the solution vector q = (η, u)τ , � is a contour in the complexk-plane, I denotes the identity matrix, ω(k) is the dispersion relation in the formof a diagonal matrix, and ρ(k) is a function of the complex spectral parameter k,defined in terms of the prescribed initial and boundary conditions, and independentof x or t .

Remark 1.1. For scalar evolution PDEs the analogue of Equation (1.5) can beconstructed in at least three different ways, by:

(i) Performing the simultaneous spectral analysis of the associated Lax pair orequivalently the spectral analysis of a certain differential 1-form [8];

(ii) Using the Fourier transform with respect to x to express q(x, t) as as inte-gral along the real k-axis and then deforming the integration contour usingCauchy’s theorem [9];

(iii) Using a reformulation of Green’s theorem [16].

For scalar PDEs the approach (ii) is the simplest, while for systems of PDEsit appears that the most convenient approach is (i). Hence, since the linearisedKdV–KdV system can be decoupled to two scalar PDEs (see Section 2), we willuse (ii) for the construction of an integral representation of the KdV–KdV system,while we will use (i) for the analogous construction for the linearised classicalBoussinesq system.

Remark 1.2. To minimise technicalities, we assume throughout this paper thatall prescribed initial and boundary conditions are smooth functions, but it is possi-ble to extend our results to the case of less regular functions.

1.1. STRUCTURE OF THE PAPER

In Section 2 we present an alternative formulation of the linearised KdV–KdVsystem and of the linearised classical Boussinesq system. These alternative for-mulations provide the starting point of the subsequent analysis. The algorithmic


derivation of such formulations for a large class of systems of PDEs is presentedin the Appendix.

In Section 3 we consider boundary value problems for KdV–KdV on the halfline: we start by reviewing the relevant results for the scalar linearised KdV equa-tion, then we prove the well posedness result for the system, and also give theexplicit integral representation for its solution.

In Section 4 similar results are derived for KdV–KdV on a finite interval [0, L],with emphasis on the characterisation of the boundary conditions that yield a wellposed problem.

In Sections 5 and 6 we study the boundary value problem on the half line andon a finite interval for the classical Boussinesq system, respectively, deriving thewell posedness results and giving the integral representation of the solution. Thedetails of the derivation of the integral representation, which as mentioned aboveis based on the spectral analysis of a differential 1-form (see Equation (2.16)) arepresented in the Appendix. Finally, in Section 7, we indicate how to extend theabove results to nonhomogeneous linear systems. By treating nonlinear systemsas a forced version of the linear ones, this allows us to extend the well-posednessresults to the nonlinear case, at least for small times.

1.2. NOTATION

• Boundary conditions on the left (x = 0) and on the right (x = L) will bedenoted by f and g, respectively.

• Functions of k representing x- and t-transforms will be indicated by super-scripts ˆ and ˜, respectively. E.g., U0(x) → U0(k) and f (t) → f (k).

• D+ and D− denote the domains in the upper and lower half complex k-planewhere Re ω(k) < 0, with ω(k) the underlying dispersion relation. When thereis a risk of confusion, D±(U) will denote the relevant domains associated withthe equation satisfied by U(x, t).

• S(R) denotes the space of Schwartz functions on the half line [0, ∞).

2. An Alternative Representation of the Linearised Systems

2.1. LINEARISED KDV–KDV SYSTEM

It is straightforward to reduce this system to two independent scalar PDEs. Indeed,by setting U = η + u and V = η − u, the linearised KdV–KdV system (1.3)decouples to

Ut + Ux + Uxxx = 0, (2.1a)

Vt − Vx − Vxxx = 0. (2.1b)

It can be verified that Equation (2.1a) is equivalent to the equation

(e−ikx+ω(k)tU)t + (e−ikx+ω(k)tXU)x = 0, k ∈ C, (2.2)


where the dispersion relation ω(k) and the function XU(x, t, k) are given by

ω(k) = i(k − k3), XU = (1 − k2)U + ikUx + Uxx. (2.3)

Suppose that Equation (2.1a) is valid in a simply connected domain D ⊂ R2 with

a smooth boundary ∂D . Equation (2.2) and Green’s theorem imply∫∂D

e−ikx+ω(k)t (U dx − XU dt) = 0. (2.4)

Similarly, Equation (2.1b) is equivalent to the equation

(e−ikx+ω(−k)tV )t + (e−ikx+ω(−k)tXV )x = 0, k ∈ C, (2.5)

where

XV = −(1 − k2)V − ikVx − Vxx, (2.6)

yielding∫∂D

e−ikx+ω(−k)t (V dx − XV dt) = 0. (2.7)

Equations (2.4) and (2.7), as well as the equations obtained from these by usingcertain transformations in the complex k-plane, play a crucial role in the analysis ofthe well-posedness of the linear KdV–KdV system. We will refer to these equationsas the global relations.

Equation (2.2) provides the starting point for constructing an integral repre-sentation of the solution U(x, t). For scalar evolution equations, such as Equa-tions (2.1), the easiest way to achieve this is to use the classical Fourier transformand a certain contour deformation, see [9]. An alternative approach is to use thesimultaneous spectral analysis of the associated Lax pair [15]. The Lax pair asso-ciated with Equation (2.1a) follows immediately from Equation (2.2): this equationimplies that there exists an auxiliary function M(x, t, k) such that

e−ikx+ω(k)tU = Mx, e−ikx+ω(k)tXU = −Mt.

Letting µ(x, t, k) = eikx−ω(k)tM(x, t, k), these equations become the Lax pair

µx − ikµ = U, µt + ω(k)µ = −XU. (2.8)

It is convenient to rewrite these two equations as the single equation

d[e−ikx+ω(k)tµ] = e−ikx+ω(k)t (U dx − XU dt). (2.9)

Similar results hold for the solution V (x, t) of (2.1b).

Remark 2.1. Equation (2.2) is equivalent to the statement that the differential1-form WU defined by

WU = e−ikx+ω(k)t [U(x, t) dx − XU(x, t, k) dt], (2.10)

is a closed form. For a simply connected domain, this immediately implies Equa-tion (2.4). Furthermore, for such a domain a closed form is also exact, hence thereexists a 0-form M such that dM = WU , which immediately implies (2.8).


2.2. LINEARISED CLASSICAL BOUSSINESQ SYSTEM

It can be verified that the linearised classical Boussinesq system (1.4) is equivalentto the equation

(e− i2 (k− 1

k)x+ω(k)t q(x, t, k))t − (e− i

2 (k− 1k)x+ω(k)tX(x, t, k))x = 0, (2.11)

where

ω(k) = ik − 1

k

k + 1k

, q(x, t, k) = η − iux + 1

ku, (2.12)

and

X(x, t, k) = 2

k + 1k

(uxt − η − 1

k(iut + u)

). (2.13)

Suppose that the system (1.4) is valid in a simply connected domain D ⊂ R2 with

a smooth boundary ∂D . Equation (2.11) and Green’s theorem imply the globalrelation

∫∂D

e− i2 (k− 1

k)x+ω(k)t (q dx + X dt) = 0, (2.14)

as well as the Lax pair

µx − i

2

(k − 1

k

)µ = η − iux + u

k,

µt + ik − 1

k

k + 1k

µ = 2

k + 1k

(uxt − η − 1

k(iut + u)

).

(2.15)

Remark 2.2. This formulation is equivalent to the statement that the differential1-form

W = e− i

2 (k− 1k)x+i

k− 1k

k+ 1k

t ×

×[(

η − iux + u

k

)dx + 2

k + 1k

(uxt − η − 1

k(iut + u)

)dt

](2.16)

is a closed form. See also Remark 2.1.


3. The Half-Line Problem for the Linear KdV–KdV System

3.1. WELL POSEDNESS RESULTS

3.1.1. Well Posedness for Equation (2.1a)

Let U0(k) denote the x-Fourier transform of U0(x)

U0(k) =∫ ∞

0e−ikxU0(x) dx, Im k � 0, (3.1)

and let f Uj (k) denote the t-transform of ∂

jx U(0, t) given by

f Uj (k) =

∫ T

0eω(k)s∂j

x U(0, s) ds, j = 0, 1, 2, k ∈ C. (3.2)

Substituting the definition of XU (see Equation (2.3)) in the global relation (2.4),and noting that

∂D = {t = 0, 0 � x < ∞}∪ {x = 0, 0 � t � T } ∪ {t = T , 0 � x < ∞}, (3.3)

we find

f U2 (k) + ikf U

1 (k) + (1 − k2)f U0 (k)

= −U0(k) + eω(k)T UT (k), Im k � 0, (3.4)

where UT (k) denotes the x-Fourier transform of U(x, T ), i.e.

UT (k) =∫ ∞

0e−ikxU(x, T ) dx, Imk � 0. (3.5)

The global relation (3.4) can be used to identify the number of boundary conditionsthat must be specified at x = 0 in order for the problem to be well posed. In thisrespect we note:

(i) U0(k) and UT (k) are defined for Im k � 0, while f Uj (k) are entire functions

of k. Hence Equation (3.4) is valid for Im k � 0.(ii) The functions f U

j (k) depend on k only through ω(k). It follows that thesefunctions are invariant under any transformation k → λ(k) which leaves ω(k)

invariant. The equation ω(k) = ω(λ) has three roots, the root λ = k and thetwo roots λ1(k), λ2(k) which solve the quadratic equation

λ2 + λk + k2 − 1 = 0. (3.6)

We distinguish these roots by fixing their asymptotic behaviour:

λ1 ∼ e2πi

3 k, λ2 ∼ e4πi

3 k, k −→ ∞. (3.7)

(iii) U(x, t) satisfies an evolution equation, thus for t < T , U(x, t) cannot dependon U(x, T ).


Figure 1. The domains D+ and D− for Equation (2.1a).

It follows that UT should not contribute to the representation of the solution. A pos-sible argument to eliminate UT from the expression (3.4) is to assume that theboundary conditions are such that U(x, t) decays for large t . In this case, lettingT → ∞, and restricting k to the domain where Re ω(k) � 0, we see that theterm in Equation (3.4) involving UT vanishes. We note that if T → ∞, thenthe functions f U

j (k) are not entire functions, but they are still well defined forRe ω(k) � 0. Let D+ and D− denote the domains in the upper and lower complexk-plane where Re ω(k) < 0. If ω(k) is given by Equation (2.3), these domains aredepicted in Figure 1.

Ignoring the UT (k) term, Equation (3.4) becomes

f U2 (k) + ikf U

1 (k) + (1 − k2)f U0 (k) = −U0(k), k ∈ D−. (3.8)

Using the fact that the transform k → λ(k) maps D+ ∪D− to itself, it follows from(3.7) that if k ∈ D+ then λ1 ∈ D−

1 and λ2 ∈ D−2 .

Combining the above three observations it is now straightforward to determinethe number of boundary conditions needed at x = 0. Since Equation (3.8) is validin D− = D−

1 ∪D−2 , we can replace k by λ1(k) and by λ2(k) to obtain two equations,

both of which are valid for k in D+, namely

f U2 (k) + iλ1f

U1 (k) + (1 − λ2

1)fU0 (k) = −U0(λ1),

f U2 (k) + iλ2f

U1 (k) + (1 − λ2

2)fU0 (k) = −U0(λ2), k ∈ D+.

(3.9)

These are two equations for the three functions f Uj , j = 0, 1, 2. Thus for a well

posed problem one needs to specify one of these functions, i.e. one boundarycondition at x = 0.

Remark 3.1. It will be shown in Section 3.2 that the assumption that U(x, t)

decays as t → ∞ can be relaxed.


3.1.2. Well Posedness for Equation (2.1b)

In analogy with Equations (3.1), (3.2) we set

V0(k) =∫ ∞

0e−ikxV0(x) dx, Im k � 0, (3.10)

f Vj (k) =

∫ T

0eω(−k)s∂j

x V (0, s) ds, j = 0, 1, 2, k ∈ C. (3.11)

The global relation (2.7) takes the form

f V2 (k) + ikf V

1 (k) + (1 − k2)f V0 (k) = V0(k), k ∈ D−. (3.12)

The roots λ1 and λ2 which leave the dispersion relation invariant are still definedby Equations (3.6)–(3.7), however the domains D+, D− are now different, as ω isnow evaluated at −k (see Figure 2).

If k ∈ D+1 then λ2 ∈ D−, and if k ∈ D+

2 , then λ1 ∈ D−. Thus evaluatingEquation (3.12) at λ1(k) yields a single equation valid for k in D+

2 , and evaluatingEquation (3.12) at λ2(k) yields a single equation for k in D+

1 . Hence we haveonly one equation valid in D+, and in order to define a well posed problem weneed to prescribe two boundary conditions at x = 0. Once this is done, two ofthe functions fj are known and the third one is obtained by solving in D+ thefollowing equations:

k ∈ D+1 : f V

2 (k) + iλ2fV1 (k) + (1 − λ2

2)fV0 (k) = V0(λ2), (3.13)

k ∈ D+2 : f V

2 (k) + iλ1fV1 (k) + (1 − λ2

1)fV0 (k) = V0(λ1). (3.14)

Figure 2. The domains D+ and D− for Equation (2.1b).


3.1.3. Well Posedness for the Linear KdV–KdV System

For k ∈ D+(U) (defined in terms of the dispersion relation ω(k) given by (2.3)and depicted in Figure 1), Equations (3.9) are valid. Also, since the global relation(3.12) is valid for k ∈ D−(V ), using the transformation k → −k and the fact thatD−(V ) = −D+(U), Equation (3.12) becomes

f V2 (−k) − ikf V

1 (−k) + (1 − k2)f V0 (−k) = V0(−k), k ∈ D+(U). (3.15)

Thus for k ∈ D+(U) there exist three algebraic Equations, (3.9) and (3.15), relatingthe boundary values of U(x, t) and V (x, t).

Recall that U = η + u, V = η − u. Introduce the following functions:

fη

j =∫ T

0eω(k)s∂j

x η(0, s) ds, f uj =

∫ T

0eω(k)s∂j

x u(0, s) ds, j = 0, 1, 2.

(3.16)

Since ω(−k) = −ω(k), we find

f Uj (k) = f

η

j (k) + f uj (k), f V

j (−k) = fη

j (k) − f uj (k). (3.17)

Note that, for any finite T , the functions defined by Equations (3.16) are entirefunctions of k, and if T → ∞, these functions are well defined for k ∈ D+.

In terms of the functions (3.16), the three Equations (3.9) and (3.15) yield, fork ∈ D+(U):

(fη

2 + f u2 ) + iλ1(k)(f

η

1 + f u1 ) + (1 − λ1(k)2)(f

η

0 + f u0 )

= −(η0 + u0)(λ1(k)),

(fη

2 + f u2 ) + iλ2(k)(f

η

1 + f u1 ) + (1 − λ2(k)2)(f

η

0 + f u0 )

(3.18)= −(η0 + u0)(λ2(k)),

(fη

2 − f u2 ) − ik(f

η

1 − f u1 ) + (1 − k2)(f

η

0 − f u0 )

= (η0 − u0)(−k).

This is a nonsingular system of three equations involving the six functions fη

0 ,f

η

1 , fη

2 , f u0 , f u

1 , f u2 . These functions are the t-transforms of the boundary values

η(0, t), ηx(0, t), ηxx(0, t), u(0, t), ux(0, t), uxx(0, t). Thus for a problem to be wellposed we need to prescribe three of these functions, i.e. three boundary conditionsat x = 0.

We summarise this result in the following proposition.

PROPOSITION 3.1. Consider the linear KdV–KdV system (1.3) posed for x ∈(0, ∞) and t ∈ (0, T ), with prescribed initial conditions

η(x, 0) = η0(x) ∈ S(R+), u(x, 0) = u0(x) ∈ S(R+).

If three conditions fi(t) ∈ C∞[0, T ], i = 1, 2, 3 are prescribed at x = 0 forany three of the functions η(0, t), ηx(0, t), ηxx(0, t), u(0, t), ux(0, t), uxx(0, t)

(compatible with the given initial conditions at t = 0), the resulting boundaryvalue problem is well posed.


3.2. THE INTEGRAL REPRESENTATION OF THE SOLUTION

3.2.1. Integral Representation for the Solution of (2.1a)

It was shown in [15] that U(x, t) admits the integral representation

U(x, t)

= 1

2π

{∫ ∞

−∞eikx−ω(k)t U0(k) dk +

∫∂D+(U)

eikx−ω(k)t f U (k) dk

}, (3.19)

where ω(k) and U0(k) are given by Equations (2.3) and (3.1), ∂D+ is the orientedboundary of the domain D+(U), given explicitly by

∂D+ = {k = k1 + ik2 ∈ C: k2 > 0, 3k21 − k2

2 − 1 = 0}∪

{k: k2 = 0, |k1| � 1√

3

}, (3.20)

and f U (k) is defined by

f U (k) = f U2 (k) + ikf U

1 (k) + (1 − k2)f U0 (k), (3.21)

where the functions f Uj are given by Equations (3.2). The three functions f U

j

satisfy the system (3.9) for k ∈ D+(U). Hence if one of them is prescribed, theother two are obtained by solving this system. This implies that the function f U isexplicitly known.

Remark 3.2. Using the integral representation (3.19) it is now possible to relaxthe assumption that U(x, t) decays as t → ∞. Indeed, for finite T the right-hand side of Equations (3.9) contains also the terms eω(k)T UT (λ1(k)) andeω(k)T UT (λ2(k)), respectively.

Suppose, for example, that U(0, t) = f0(t) is the given boundary condition.Hence f U

0 (k) is known and solving the system (3.19) for f U1 (k) and f U

2 (k), weobtain for f U (k) the expression

f U (k) = λ1 − k

λ2 − λ1U0(λ2) + k − λ2

λ2 − λ1U0(λ1) + (1 − k2)f U

0 (k) +

+ eω(k)T

λ2 − λ1[(k − λ1)UT (λ2) + (λ2 − k)UT (λ1)].

The important observation is that the last term does not contribute to the represen-tation of U(x, t). Indeed, its contribution to this representation involves only termsof the form∫

∂D+(U)

eikx−ω(k)(t−T )UT (λi(k) dk, i = 1, 2

and since the integrand in this expression is analytic and bounded in D+(U),Jordan’s lemma implies that these terms vanish.


3.2.2. Integral Representation for the Solution of (2.1b)

It is shown in [15] that V (x, t) admits the integral representation

V (x, t) = 1

2π

{∫ ∞

−∞eikx−ω(−k)t V0(k) dk +

∫∂D+(V )

eikx−ω(−k)t f V (k) dk

},

(3.22)

where ω(k) is as above, V0(k) is given by Equation (3.10), and ∂D+(V ) is theoriented boundary of the domain D+(V ), which in this case is given explicitly by

∂D+(V ) = {k = k1 + ik2 ∈ C: k2 > 0, 3k21 − k2

2 − 1 = 0}

∪{k: k2 = 0, |k1| � 1√

3

}, (3.23)

and f V (k) is defined by

f V (k) = f V2 (k) + ikf V

1 (k) + (1 − k2)f V0 (k), (3.24)

where the functions f Vj are given by Equations (3.11).

The function f V involves the three functions f Vj , which satisfy Equation (3.13)

for k ∈ D+2 (V ) and Equation (3.14) for k ∈ D+

1 (V ). Hence if two of thesefunctions are prescribed, the third can be computed in D+ and the function f U

is known.

Remark 3.3. As for the case of U(x, t), using the representation (3.22) it isagain possible to relax the assumption that V (x, t) decays as t → ∞.

3.2.3. Integral Representation for the Linear KdV–KdV System

Using the representations of U and V , we can obtain the representation for η and u.Equations (3.19), (3.22) together with η = (U +V )/2, u = (U −V )/2, and (3.17),imply

η(x, t) = 1

4π

{∫ ∞

−∞eikx[e−ω(k)t (η0 + u0)(k) + eω(k)t (η0 − u0)(k)] dk +

+∫

∂D+(U)

eikx−ω(k)t (f η + f u)(k) dk +

+∫

∂D+(V )

eikx+ω(k)t (fη− − f u

−)(k) dk

}(3.25)


and

u(x, t) = 1

4π

{∫ ∞

−∞eikx[e−ω(k)t (η0 + u0)(k) − eω(k)t (η0 − u0)(k)] dk +

+∫

∂D+(U)

eikx−ω(k)t (f η + f u)(k) dk −

−∫

∂D+(V )

eikx+ω(k)t (fη− − f u

−)(k) dk

}. (3.26)

The functions f η(k) and fη−(k) are defined by

f η(k) = fη

2 (k) + ikfη

1 (k) + (1 − k2)fη

0 (k),

fη−(k) = f

η

2 (−k) + ikfη

1 (−k) + (1 − k2)fη

0 (−k)(3.27)

and f u(k), f u−(k) are defined by a similar expression, with η replaced by u. Thedomains D+(U), D+(V ) are defined in Equations (3.20) and (3.23), and depictedin Figures 1 and 2. In order to derive an explicit expressions for f η(k) and f u(k)

for k in D+U ∪ D+

V = C+, we consider the following systems:

for k ∈ D+(U): use Equations (3.19);

for k ∈ D+1 (V ): λ2(k) ∈ D−(V ) → use (3.12) evaluated at λ2(k);

−k ∈ D−1 (U) → use (3.8) evaluated at −k;

−λ1(k) ∈ D−2 (U) → use (3.8) evaluated at −λ1(k);

for k ∈ D+2 (V ): λ1(k) ∈ D−(V ) → use (3.12) evaluated at λ1(k);

−k ∈ D−2 (U) → use (3.8) evaluated at −k;

−λ2(k) ∈ D−1 (U) → use (3.8) evaluated at −λ2(k).

Each of these systems is a system of three equations for six functions. Thusif three of these functions are known, the solutions of these systems gives anexpression for the remaining three, and the representations (3.25) and (3.26) areexplicit expressions involving only prescribed boundary conditions.

EXAMPLE. As an illustrative example, consider the case that the three prescribedconditions are

fη

0 (t) = η(0, t), fη

1 (t) = ηx(0, t),(3.28)

fη

2 (t) = ηxx(0, t), fη

j (t) ∈ C∞.

Hence the three functions fη

0 , fη

1 and fη

2 are known, and Equations (3.19) yield asystem for the unknown functions f u

0 , f u1 , f u

2 , k ∈ D+(U), given by

f u2 + iλ1(k)f u

1 + (1 − λ1(k)2)f u0 = −f η(λ1(k)) − (η0 + u0)(λ1(k)),

f u2 + iλ2(k)f u

1 + (1 − λ2(k)2)f u0 = −f η(λ2(k)) − (η0 + u0)(λ2(k)), (3.29)

− f u2 + ikf u

1 − (1 − k2)f u0 = −f

η−(−k) + (η0 + u0)(−k),


where f η(k), f η−(k) are defined by (3.27). Solving this system using Cramer’s rule,

we obtain

f uj = det(Bj+1)

det(C), j = 0, 1, 2,

where the matrix C is given by

C =

1 − λ21 iλ1 1

1 − λ22 iλ2 1

k2 − 1 ik −1

,

and Bj is the matrix obtained from the matrix C by replacing the j th column of C

with the right-hand side of the system (3.29). We note that the determinant of C isalways different from zero, for k ∈ D+(U). This follows from the fact that k, λ1

and λ2 are the three distinct roots of the third-order polynomial ω(k) = ω(λ).Suppose for simplicity that the given boundary conditions are all zero: f0(t) =

f1(t) = f2(t) = 0. In this case, f η(k) = fη−(k) = 0. Hence

B1 = det

(η0 + u0)(λ1) iλ1 1(η0 + u0)(λ2) iλ2 1

−(η0 − u0)(−k) ik −1

,

yielding

f u0 (k) = i

k(η0 − u0)(−k) + λ1(η0 + u0)(λ1) − λ2(η0 + u0)(λ2)

det C

and similar expressions for f u1 (k) and f u

2 (k). Hence the functions f uj (k), j =

0, 1, 2 are known for k ∈ D+(U). Similarly, using the systems described earlierwe can compute these functions for k ∈ D+

1 (V ) and k ∈ D+2 (V ). Thus all the terms

in the representations (3.25) and (3.26) are known.

Remark 3.4. If we consider the two equations for U and V separately, the aboveexample would appear not to be well posed. Indeed, the problem for U requires oneboundary conditions, while the one for V requires two boundary condition, whichcould never yield three conditions for η = U + V . This shows the importance oftreating the two problems simultaneously.


4. The Two-Point Boundary Value Problems for the Linear KdV–KdVSystem


4.1.1. Well Posedness for Equation (2.1a)

Let U0(k) denote the x-Fourier transform of U0(x) on [0, L],

U0(k) =∫ L

0e−ikxU0(x) dx, Im k � 0, (4.1)

let f Uj (k) denote the t-transform of ∂

jx U(0, t) given by (3.2), and let gU

j (k) denote

the t-transform of ∂jx U(L, t) defined by

gUj (k) =

∫ T

0eω(k)s∂j

x U(L, s) ds, j = 0, 1, 2, k ∈ C. (4.2)

Since the boundary of the domain D now also contains the segment {x = L, 0 <

t < T }, the global relation (2.4) yields

[f U2 (k) + ikf U

1 (k) + (1 − k2)f U0 (k)] − e−ikL[gU

2 (k) + ikg1U(k) +

+ (1 − k2)gU0 (k)] = U0(k) − eω(k)T UT (k), (4.3)

with

UT (k) =∫ ∞

0e−ikxU(x, T ) dx.

Since U0, UT are now entire functions, Equation (4.3) is valid for all k ∈ C. Thisequation can be supplemented with the equations obtained by the substitutionsk → λ1 and k → λ2. To obtain exponential terms with the same boundednessproperties as k → ∞, we multiply Equation (4.3) by eikL, and restrict attention to k

in the domain D+(U). Thus we obtain the following system of three equations:

eikL[f U2 (k) + ikf U

1 (k) + (1 − k2)f U0 (k)] − [gU

2 (k) + ikg1U(k) +

+ (1 − k2)gU0 (k)] = −eikLU0(k) + eω(k)T +ikLUT (k),

[f U2 (k) + iλ1f

U1 (k) + (1 − λ2

1)fU0 (k)] − e−iλ1L[gU

2 (k) + iλ1gU1 (k) +

(4.4)+ (1 − λ2

1)gU0 (k)] = −U0(λ1) + eω(k)T UT (λ1),

[f U2 (k) + iλ2f

U1 (k) + (1 − λ2

2)fU0 (k)] − e−iλ2L[gU

2 (k) + iλ2gU1 (k) +

+ (1 − λ22)g

U0 (k)] = −U0(λ2) + eω(k)T UT (λ2).

The system (4.4) involves the six functions f Uj , gU

j , j = 0, 1, 2. Anticipating that

the (unknown) terms UT will not contribute to the solution, we expect that for a


well posed problem one needs to prescribe a total of three boundary conditions.However, not all choices of three boundary conditions yield a well posed problem.Indeed, the term UT appears in three forms:

eω(k)T +ikLUT (k), eω(k)T UT (λ1), eω(k)T UT (λ2). (4.5)

Thus even if we assume that U(x, t) decays for large t and restrict k to D+(U),these terms will vanish if and only if the determinant of the left-hand side decreasesfaster than all of the terms eikL, e−iλ1L and e−iλ2L, divided by it, decay as k → ∞.It can be shown [17] that such decay is guaranteed if and only if one prescribesone boundary condition at the left endpoint x = 0 and two boundary conditions atthe right endpoint x = L. This is consistent with the fact that if L → ∞, we mustrecover the results of Section 3.

4.1.2. Well posedness for Equation (2.1b)

Let V0(k) denote the Fourier transform of V0(x) with respect to x,

V0(k) =∫ ∞

0e−ikxV (x, 0) dx,

let ∂D+V be given by (see (3.23)),

∂D−V = {k = k1 + ik2 : k2 < 0, 3k2

1 − k22 − 1 = 0}

∪{k: k2 = 0, |k1| � 1√

3

}, (4.6)

let f Vj (k), gV

j (k) be given by (3.11) and by

gVj (k) =

∫ T

0eω(−k)s∂j

x V (L, s) ds, j = 0, 1, 2, k ∈ C.

The global relation (2.7) becomes

eikL[f V2 (k) + ikf V

1 (k) + (1 − k2)f V0 (k)] −

− [gV2 (k) + ikgV

1 (k) + (1 − k2)gV0 (k)]

= eikLV0(k) − eω(−k)T +ikLVT (k), (4.7)

with

VT (k) =∫ ∞

0e−ikxV (x, T ) dx.

This equation is valid for all k ∈ C. Supplementing it with the equations obtainedby the substitutions k → λ1(k) and k → λ2(k) we obtain a system of three equa-tions involving the six functions gV

j (k), gVj (k), j = 0, 1, 2. As in the previous

case, it can be shown that this system can be effectively solved if and only iftwo boundary conditions are prescribed at x = 0 and one condition is prescribedat x = L.


4.1.3. Well Posedness for the Linear KdV–KdV System

We now combine the results of the above sections to characterise well posed bound-ary value problems for the linearised KdV–KdV system. The crucial step is thederivation of a system of equations for the twelve functions f

η

j , f uj , g

η

j , guj , j =

0, 1, 2, which appear in four Equations (4.3)–(4.4) and (4.7), as well as in the twoequations obtained from Equation (4.7) from the substitutions k → λ1(k) andk → λ2(k). After multiplying each one by the appropriate exponential in orderthat all terms are bounded in D+(U), these equations yield the following systemof six equations for the functions f U

j (k), gUj (k), f V

j (−k), gVj (−k):

eikL[f U2 + ikf U

1 + (1 − k2)f U0 ] − [gU

2 + ikgU1 + (1 − k2)gU

0 ]= −eikLU0(k) + eω(k)T +ikLUT (k),

[f U2 + iλ1f

U1 + (1 − λ2

1)fU0 ] − e−iλ1L[gU

2 + iλ1gU1 + (1 − λ2

1)gU0 ]

= −U0(λ1) + eω(k)T VT (λ1),

[f U2 + iλ2f

U1 + (1 − λ2

2)fU0 ] − e−iλ2L[gU

2 + iλ2gU1 + (1 − λ2

2)gU0 ]

= −U0(λ2) + eω(k)T VT (λ2),(4.8)[f V

2 − ikf V1 + (1 − k2)f V

0 ](−k) − eikL[gV2 − ikgV

1 + (1 − k2)gV0 ](−k)

= V0(−k) − eω(−k)T VT (−k),

e−iλ1L[f V2 − iλ1f

V1 + (1 − λ2

1)fV0 ](−k) − [gV

2 − iλ1gV1 + (1 − λ2

1)gV0 ](−k)

= e−iλ1LV0(−λ1) − e−iλ1L+ω(−k)T VT (−λ1),

e−iλ2L[f V2 − iλ2f

V1 + (1 − λ2

2)fV0 ](−k) − [gV

2 − iλ2gV1 + (1 − λ2

2)gV0 ](−k)

= e−iλ2LV0(−λ2) − e−iλ2L+ω(−k)T VT (−λ2).

Since f U (k) = f η(k) + f u(k) and f V (−k) = f η(k) − f u(k), this is a systemof six equations for the twelve functions f

η

j , f uj ,gη

j ,guj , j = 0, 1, 2. We expect

that the system will have a unique solution if six of these functions are prescribed.However, as in [17], it can be shown that not all choices yield a system whichhas the property that the terms UT and VT do not contribute to the solution of theproblem. The system (4.8) has the same structure as the system considered in [17]for the case of a scalar equation of sixth order, thus the proof presented in [17]implies that the only possible choice is to prescribe three conditions at each end.

We summarise the above discussion in the following proposition.

PROPOSITION 4.1. Consider the linearised KdV–KdV system (1.3), 0 < x < L,0 < t < T , with the initial conditions η(x, 0) = η(x) and u(x, 0) = u(x).An initial boundary value problem admits a unique solution if and only if threeconditions for either η(x, t) or u(x, t) or their spatial derivatives are prescribedat both x = 0 and x = L. This result holds assuming that the initial and boundaryconditions are sufficiently smooth and that they are compatible at x = 0, t = 0 andat x = L, t = 0.


5. The Half-Line Problem for the Linear Boussinesq System

We start by analysing the global relation (2.14) in order to characterise the wellposed problems. We then present the integral representation for the solution of thelinearised classical Boussinesq system. The derivation of this representation can befound in the Appendix.


Let η0(k), u0(k), u′0(k) be the x-Fourier transforms of the initial data η0(x), u0(x),

u′0(x), given by

η0(k) =∫ ∞

0e− i

2 (k− 1k)xη0(x) dx, (5.1)

and u0(k), u′0(k) given by a similar formula, with η0(x) replaced by u0(x), u′

0(x),respectively. Let f (k) be the t-transform of the boundary data defined by

f (k) = −∫ T

0eω(k)s

(uxt (0, s) − η(0, s) − 1

k(iut (0, s) + u(0, s))

)ds, (5.2)

where

ω(k) = ik − 1

k

k + 1k

. (5.3)

Using the definition (2.13) of X(x, t, k), since ∂D is given by (3.3), the globalrelation (8.10) becomes

2k

1 + k2f (k) = −

(η0 − iu′

0 + u0

k

)(k) +

+ eω(k)T

(ηT − iu′

T + uT

k

)(k), k ∈ C

−, (5.4)

where ηT (k), uT (k), u′T (k) are defined by

ηT (k) =∫ ∞

0e− i

2 (k− 1k)xη(x, T ) dx,

(5.5)

uT (k) =∫ ∞

0e− i

2 (k− 1k)xu(x, T ) dx,

and u′T (k) has a similar definition with u(x, t) replaced by u′(x, T ).

The global relation can be used to identify the number of boundary conditionsneeded in order to define a well posed problem. Indeed, in this case the dispersionrelation ω(k) is invariant under the transformation k → −k. An argument similarto the one used in (iii), Section 3.1, implies that for k ∈ D−, where D− denotes thethird quadrant of the complex k-plane, the global relation must hold without the


terms ηT (k), uT (k), u′T (k):

2k

1 + k2f (k) = −

(η0 − iu′

0 + u0

k

)(k), k ∈ D−. (5.6)

Let D+ denote the first quadrant of the complex k plane; then D− = −D+, and ifk ∈ D+ then −k ∈ D−, and we deduce that Equation (5.6) yields the followingequation valid in D+:

f (−k) = 1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k

). (5.7)

The boundary data appearing in Equations (5.6)–(5.7) are simply related: u(0, t)

and ut(0, t) can be computed from each other by integration or differentiation,while uxt (0, t) and η(0, t) are related by the equation ux(0, t) = −ηt(0, t). Thusgiven one boundary condition for u(0, t), or for η(0, t), (5.6) is one equation for theremaining boundary data valid in D−, and (5.7) is one equation for the remainingboundary data valid in D+.

We summarise the above discussion in the following proposition. More detailsare given in Theorem 5.1.

PROPOSITION 5.1. Consider the linearised Boussinesq system (1.4), for 0 �x < ∞, 0 � t � T . Let the initial conditions η(x, 0) = η0(x) and u(x, 0) = u0(x)

be given. If either η(0, t) or any one of the functions ∂jx u(0, t), j = 0, 1, is given,

the resulting initial boundary value problem admits a unique smooth solution,provided that the initial and boundary conditions are sufficiently smooth, decayas x → ∞, and are compatible at x = t = 0.

5.2. THE INTEGRAL REPRESENTATION OF THE SOLUTION

We give in the next proposition the expression for the integral representation of thesolution of a well posed boundary value problem for the linearised classical Boussi-nesq system, posed on the half-line. The functions involved in this representationare defined in the previous section. The proof is given in the Appendix.

PROPOSITION 5.2. Assume that a given boundary value problem for the linearclassical Boussinesq system (1.4) has a unique solution (η(x, t), u(x, t))τ for 0 <

x < ∞, 0 < t < T which is sufficiently smooth and has sufficient decay asx → ∞, uniformly in 0 < t < T . Then this solution can be represented by

η(x, t) = η(x, 0) −∫ t

0ux(x, s) ds, (5.8)

u(x, t) = 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

(η0 − iu′

0 + u0

k

)dk

k+

+ 2∫

∂D+e

i2 (k− 1

k)x−ω(k)t f (k)

1 + k2dk

}, (5.9)


where D+ denotes the first quadrant of the complex k-plane. In these expressions,p.v. indicates that the integrals must be interpreted as principal value at k = 0.

The definition of f (k) implies that

f (−k) = −∫ T

0eω(k)s

(uxt − η + 1

k(iut + u)

)(0, s) ds.

The expressions for f (k) and f (−k) can be combined and yield the following twoequations:

f (k) = f (−k) + 2

k

∫ T

0eω(k)t (iut + u)(0, t) dt, (5.10)

f (k) = −f (−k) − 2∫ T

0eω(k)t (uxt − η)(0, t) dt. (5.11)

For k ∈ D+, f (−k) is given explicitly by Equation (5.7) in terms of the initialconditions. Thus given one boundary condition for u(0, t), or for η(0, t), eitherEquation (5.10) (if the prescribed boundary condition is u(0, t)) or Equation (5.11)(if the prescribed boundary condition is either ux(0, t) or η(0, t)) yields f (k) interms of the given data only. This implies that the expressions (5.8), (5.9) containonly known terms.

Remark 5.1. The zeros k = ±i of the function 1 + k2, which appears in thedenominator in the last integral of the representation formula (5.9), do not poseany problem as limk→±i e−ω(k)t = 0.

Remark 5.2. The linearised classical Boussinesq system yields the second orderlinear equation

utt − uxx − uxxtt = 0.

Looking for solutions of the form exp( i2(k− 1

k)±ω(k)t), we find that ω(k) is given

by Equation (5.3). Thus we expect that both of the above exponentials appear inthe spectral representation of u. This is indeed the case; although the representation(5.9) contains only the exponential of −ω(k), the exponential of ω(k) appearswhen one computes explicitly the principal value integral.

5.3. EXISTENCE AND UNIQUENESS RESULTS

In this section we prove the existence and uniqueness of the solution of sometypical well-posed boundary value problems for this equation.


THEOREM 5.1. Consider the linearised classical Boussinesq system (1.4) for0 � x < ∞, 0 � t � T , with the initial conditions η(x, 0) = η0(x), u(x, 0) =u0(x) and the boundary condition u(0, t) = f0(t). Assume that the given functionsare sufficiently smooth, and have sufficient decay as x → ∞. Also assume thatthey are compatible at x = t = 0. Then the above initial boundary value problemhas a unique smooth solution which decays as x → ∞, uniformly in 0 < t < T .This solution is given by (5.8)–(5.9), where the function f (k) is given by

f (k) = 1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k

)+

+ 2

k

∫ T

0eω(k)t{f0(t) + if ′

0(t)} dt, (5.12)

and η0, u0, u′0 are defined by Equation (5.1), and f ′

0(t) = d f0(t) / dt .Proof. The verification of formula (5.12) follows immediately from Equa-

tion (5.10). To prove the theorem, we need to show that the functions η(x, t) andu(x, t) defined by Equations (5.8) and (5.9) satisfy Equations (1.4) as well as theinitial and the boundary conditions.

Since the x and t dependence of the relevant functions is exponential, it is easyto verify that these functions satisfy the system. Indeed, this is a straightforwardverification using the expressions (5.9) for u and the alternative explicit expression(8.30) for η, both derived in the Appendix.

For u(x, 0), using the definition (5.1) for u(k), and computing explicitly the firstprincipal value integral in the representation (5.9), we find

u(x, 0) = 1

4π

{−

∫ ∞

−∞e

i2 (k− 1

k)x

(k + 1

k

)u0(k)

dk

k+

+ p.v.

∫∂D+

ei2 (k− 1

k)x 2f (k)

1 + k2dk

}.

The integrand in the second integral is analytic in D+; thus this integral vanishes.Since d(k − 1

k) = 1 + 1/k2 the first integral yields u(x). Similarly for η(x, 0).

Finally, we verify that the functions η(x, t) and u(x, t) given by the expressions(5.8) and (5.9) satisfy the boundary conditions. Set x = 0 in the expression (5.9);then the integral is well defined with no need for a principal value, and usingEquation (5.10), we obtain

u(0, t) = 1

4π

{∫ ∞

−∞e−ω(k)t

(η0 − iu′

0 + u0

k

)dk

k+

+∫

∂D+e−ω(k)t

[η0(−k) − iu′

0(−k) − u0(−k)

k+

+ 4

1 + k2

∫ T

0eω(k)s(u + iut )(0, s) ds

]dk

k

}.


Since the terms e−ω(k)t η, e−ω(k)t ux , and e−ω(k)t u/k are analytic in the second quad-rant, the integral over ∂D+ of these terms can be deformed to an integral along thereal line. Using the fact that ω(k) = ω(−k), and replacing k with −k in the secondintegral over the real line, it follows that all terms containing the initial data η0 andu0 vanish. Computing explicitly the remaining term containing the boundary datau(0, t) = f0(t), using again analyticity, we find

u(0, t) = 1

4π

∫ ∞

−∞e−ω(k)t

∫ t

0eω(k)s(f0 + if ′

0)(s) dsdk

1 + k2

= 1

4π

∫ ∞

−∞e−ω(k)t

∫ t

0eω(k)sf0(s) ds

4(1 + ω(k))

k(1 + k2)dk

= 1

2π

∫ ∞

−∞e−ω(k)t

[∫ t

0eω(k)sf0(s) ds

]dω(k).

Thus u(0, t) = f0(t). �Remark 5.3. If the boundary condition in Theorem 5.1. is replaced by either of

the conditions

(a) ux(0, t) = f1(t), or (b) η(0, t) = g0(t),

then the function f (k) is given respectively by

(a) f (k) = −1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k

)−

− 2∫ T

0eω(k)t

(f ′

1(t) + η1(t))

dt,

η1(t) = η0(0) −∫ t

0f1(s) ds,

or

(b) f (k) = −1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k

)−

− 2∫ T

0eω(k)t (g0(t) + g′′

0 (t)) dt.


6. Two-Point Boundary Value Problem for the Linear Boussinesq System

6.1. WELL-POSEDNESS RESULTS

Let η0(k), u0(k), u′0(k) be the x-Fourier transforms of the initial data η0(x), u0(x),

u′0(x), given by

η0(k) =∫ L

0e− i

2 (k− 1k)xη0(x) dx, (6.1)

and u0(k), u′0(k) given by a similar formula, with η0(x) replaced by u0(x), u′

0(x),respectively. Let f (k) and g(k) be the t-transforms of the boundary data definedby

f (k) = −∫ T

0eω(k)t

(uxt (0, t) − η(0, t) − 1

k(u(0, t) + iut (0, t))

)dt,

(6.2)

g(k) = −∫ T

0eω(k)t

(uxt (L, t) − η(L, t) − 1

k(u(L, t) + iut (L, t))

)dt.

The global relation (2.14) becomes

f (k) − e− i2 (k− 1

k)Lg(k) = −1 + k2

2k

(η0 − iu′

0 + u0

k

)+

+ eω(k)T 1 + k2

2k

(ηT − iu′

T + uT

k

), k ∈ C,

(6.3)

where ηT , uT , u′T are defined in (5.5).

Since the functions appearing on the right-hand side of (6.3) are now entirefunctions, this equation is valid for all k ∈ C. In addition, it can be supplementedwith the equation obtained by replacing k with −k:

f (−k) − ei2 (k− 1

k)Lg(−k) = 1 + k2

2k

(η0 − iu′

0 − u0

k

)−

− eω(k)T 1 + k2

2k

(ηT − iu′

T − uT

k

), k ∈ C.

(6.4)

Equations (6.3), (6.4) involve eight boundary data. However these boundary dataare related. Let w = 0 or w = L. Then u(w, t) and ut(w, t) can be computedfrom each other by integration or differentiation, while uxt (w, t) and η(w, t) arerelated by the equation ux(∗, t) = −ηt(∗, t). Hence these two equations containfour unknow boundary terms. Anticipating that the (unknown) terms ηT , uT , u′

T

will not contribute to the solution, we expect that for a well posed problem oneneeds to prescribe a total of two boundary conditions. However, not all choices of


two boundary conditions yield a well posed problem: the analysis of the conditionsnecessary for the terms computed at t = T to give no contribution to the finalrepresentation implies that we need to give one condition at each end. In addi-tion, not all conditions can be chosen. To identify the correct boundary conditions,define

f η(k) =∫ T

0eω(k)t (η + ηtt )(0, t) dt,

gη(k) =∫ T

0eω(k)t (η + ηtt )(L, t) dt,

f u(k) =∫ T

0eω(k)t (u + iut )(0, t) dt,

gu(k) =∫ T

0eω(k)t (u + iut )(L, t) dt.

By definition, since uxt = −ηtt , the functions f (k), g(k) defined in (6.2) are givenby

f (k) = f η(k) + 1

kf u(k), g(k) = gη(k) + 1

kgu(k).

In addition,

f η(−k) = f η(k), gη(−k) = gη(k),(6.5)

f u(−k) = f u(k), gu(−k) = gu(k).

We assume that two boundary conditions are prescribed in such a way that twoof the four functions f η(k), f u(k), gη(k), gu(k) are known. We must show that theremaining two can be computed from the known data. For concreteness, assumethat the given data are u(0, t) and u(L, t); the other cases can be treated similarly.Then f u(k) and gu(k) are known, and we need to compute f η(k), gη(k). Equations(6.3), (6.4) yield the system

(e

i2 (k− 1

k)L −1

1 −ei2 (k− 1

k)L

) (f η(k)

gη(k)

)=

(F1(k) + T1(k)

F2(k) + T2(k)

), (6.6)

where

F1(k) = 1

k

[−e

i2 (k− 1

k)Lf u(k) + gu(k) −

− ei2 (k− 1

k)L 1 + k2

2

(η0(k) − iu′

0(k) + u0(k)

k

)],

T1(k) = 1 + k2

2keω(k)T + 1

2k− 1

k

k+ 1k

L(

ηT (k) − iu′T (k) + uT (k)

k

),


F2(k) = 1

k

[f u(k) − e

i2 (k− 1

k)Lgu(k) +

+ 1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k

)],

T2(k) = −1 + k2

2keω(k)T

(ηT (−k) − iu′

T (−k) − uT (−k)

k

).

The determinant D(k) of the system (6.6) is given by

D(k) = 1 − ei(k− 1k)L. (6.7)

Hence by Cramer’s rule

D(k)f η(k) = det

(F1(k) + T1(k) −1F2(k) + T2(k) −e

i2 (k− 1

k)L

),

D(k)gη(k) = det

(e

i2 (k− 1

k)L F1(k) + T1(k)

1 F2(k) + T2(k)

),

and computing these terms explicitly

D(k)f η(k) = −ei2 (k− 1

k)L(F1(k) + T1(k)) + F2(k) + T2(k) + D(k)f u(k),

D(k)gη(k) = ei2 (k− 1

k)L(F2(k) + T2(k)) − (F1(k) + T1(k)) + D(k)gu(k).

This system is nonsingular except at the zeros {kh} of D(k), given by kh = hπL

±√(hπ

L)2 + 1. The terms T1(k), T2(k) do not contribute to the representation of the

solution except for the contribution of the resulting poles. Hence the solution ofthis system yields the explicit expression for f (k), g(k) in terms of an integralsupplemented, in general, by a discrete sum. Analogous formulas hold for anyother sets of boundary conditions from which any two of the four functions f η(k),f u(k), gη(k), gu(k) can be computed. In summary, we have the following result.

PROPOSITION 6.1. Consider the linearised classical Boussinesq system (1.4),for 0 < x < L, 0 < t < T . Let the initial conditions η(x, 0) = η(x) andu(x, 0) = u(x) and either η(w, t) or any one of the functions ∂

jx u(w, t), j = 0, 1,

be given for both w = 0 and w = L. The resulting initial boundary value problemis well posed, provided that the initial and boundary conditions are sufficientlysmooth and that they are compatible at x = 0, t = 0 and at x = L, t = 0.

6.2. THE INTEGRAL REPRESENTATION

For completeness, we give the expression for the integral representation of thesolution of a well posed two-point boundary value problem. The proof follows thesame lines as the proof for the half-line case given in the previous section, usingthe well posedness results valid in this case.


PROPOSITION 6.2. Consider the linearised classical Boussinesq system (1.4),for 0 < x < L, 0 < t < T , and assume that there exists a unique solution whichis sufficiently smooth. Then this solution can be represented in the form

η(x, t) = η0(x) −∫ t

0ux(x, s) ds, (6.8)

u(x, t) = 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

(η0 − iu′

0 + u0

k

)dk

k+

+ 2∫

∂D+e

i2 (k− 1

k)x−ω(k)t f (k)

1 + k2dk +

+ 2∫

∂D−e

i2 (k− 1

k)(x−L)−ω(k)t g(k)

1 + k2dk

}. (6.9)

Here the function ω(k) is defined in (5.3), the regions D+ and D− denote the firstand third quadrants of the complex k-plane, respectively, the functions η0(k), u0(k),u′

0(k) are defined by (6.1), and f (k), g(k) are given explicitly in terms of the givenboundary conditions.

Remark 6.1. If the homogeneous boundary conditions u(0, t) = u(L, t) = 0are given, the representation should involve only a discrete sum. Indeed, the systemand the given initial conditions for u(x, 0) and η(x, 0) yield for the function u(x, t)

the second order PDE

utt − uxx − uxxtt = 0, u(x, 0) = u0(x), ut (x, 0) = u1(x).

This equation, with given homogeneous Dirichlet boundary conditions, can besolved by the sine transform. Setting

u = 2

L

∫ L

0u(x, t) sin

(nπx

L

)dx,

we obtain the equation

utt + k(n)2

1 + k(n)2u = 0,

where k(n) = nπ/L. The solution is then

u = Anek(n)√

1+k(n)2t + Bne

− k(n)√1+k(n)2

t,

where

An = 1

2

(u0 −

√1 + k(n)2

k(n)u1

), Bn = 1

2

(u0 +

√1 + k(n)2

k(n)u1

)


depend on the prescribed initial conditions u(x, 0) and ut(x, 0). The solution u(x, t)

of the original initial boundary value problem can hence be represented as

u(x, t) =∑n∈

sin

(k(n)πx

L

)(Ane

k(n)√1+k(n)2

t + Bne− k(n)√

1+k(n)2t).

This representation can be obtained from the integral representation (6.9) by mak-ing the change of variable k → 1/2(k − 1/k), computing explicitly the principalvalue integral and the contour integrals, and using the residue theorem to evaluatethe contribution due to the zeros of the determinant (k). The relevant computationis similar to the one carried out in detail in [14].

7. The Nonlinear Systems

The study of the linearised system associated with a given system can be used toidentify the boundary conditions for which the nonlinear system is linearly wellposed. Treating the nonlinear term as a forcing term, it is possible to prove thateach of these boundary value problems is well posed at least locally in time (orfor given data sufficinetly small in an appropriate norm). Numerical investigationscan then be used to verify this assertion and to compute the unique approximatesolution of any such problem.

We now sketch this approach in the case of the classical Boussinesq system.Similar considerations apply for the KdV–KdV system.

7.1. THE FORCED LINEARISED BOUSSINESQ SYSTEM ON THE HALF-LINE

The forced linearised system is

ηt + ux = f1(x, t),

ut + ηx − uxxt = f2(x, t).(7.1)

For this problem, a result similar to Proposition 5.1 is valid, but with the followingmodifications.

Let

F(k, t) =∫ t

0

∫ ∞

0e− i

2 (k− 1k)x+ω(k)s

[f1(x, s) + 1

2

(k + 1

k

)f2(x, s)

]dx ds.

Then

• The representation formula (5.8) is valid with the additional term∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)tF (k, t)

(1 + 1

k2

)dk.


• The representation formula (5.9) is valid with the additional term∫ ∞

−∞e

i2 (k− 1


dk

k.

• The global relation (5.4) becomes2k

1 + k2f (k) = −

(η0 − iu′

0 + u0

k

)+

+ eω(k)T

(ηT − iuT ,x + uT

k

)+ F(k, T ). (7.2)

It follows that, in all explicit formulas for the spectral function, such as Equa-tion (5.12), we must add the term F(k, T ). For example, given the initial conditionsη(x), u(x) and the boundary condition u0(t), the spectral function f (k) is given by

f (k) = 1 + k2

2k

(η0(−k) − iu′

0(−k) − u0(−k)

k+ F(−k, T )

)+

+ 2

k

∫ T

0eω(k)t{u0(t) + iu′

0(t)} dt.

Remark 7.1. We could simply add to the representation of η(x, t) the term∫ t

0 f1(x, s) ds. However, the formulation in terms of F(k, t) is more convenientfor deriving a local well posedness result.

7.2. THE CLASSICAL BOUSSINESQ SYSTEM

For the classical Boussinesq system the forcing is given by

f1(x, t) = (uη)x(x, t), f2(x, t) = u(x, t)ux(x, t).

Thus the functions η, u satisfying the classical Boussinesq system can be repre-sented by the following expressions:

η(x, t) = η(x, 0) −∫ t

0ux(x, s) ds −

−∫ ∞

−∞e

i2 (k− 1


(1 + 1

k2

)dk,

u(x, t) = 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

[(η − iux + u

k

)+ F(k, t)

]dk

k+

+ 2∫

∂D+e

i2 (k− 1

k)x−ω(k)t f (k)

1 + k2dk

},


where

F(k, t) =∫ t

0

∫ ∞

0e− i

2 (k− 1k)x+ω(k)s

[(uη)x(x, s) +

+ 1

2

(k + 1

k

)u(x, s)ux(x, s)

]dx ds.

Thus, if ulin denotes the solution of the corresponding linearised problem, the dif-ference u − ulin is given by an integral with respect to k, whose integrand containsthe term F(k, t), which involves an integral between 0 and t . For t sufficientlysmall, an appropriate norm of this difference is small.

8. Appendix

8.1. THE BASIC DIFFERENTIAL FORM AND ITS SPECTRAL ANALYSIS

Let the N -vector function q(x, t) satisfy the linear evolution equation

L(∂t , ∂x)q = 0, (x, t) ∈ D, (8.1)

where L is an N × N matrix linear differential operator of ∂t and ∂x with constantcoefficients, in which ∂t appears only in the first power.

This equation can be written formally as

qt + �(−i∂x)q = 0, (8.2)

where � is a matrix N × N pseudo-differential operator of −i∂x with constantcoefficients. We assume that the constant matrix �(k) can be diagonalised, so thatthere exists an invertible N × N matrix A(k) such that

�(k) = A−1(k)ω(k)A(k), ω(k) = diag(ω1(k), . . . , ωN(k)). (8.3)

The starting point of the spectral transform method approach is the observationthat Equation (8.1) can be written in the form

(e−ikxI+ω(k)tA(k)q(x, t))t − (e−ikxI+ω(k)tA(k)X(x, t, k))x = 0, (8.4)

where I is the identity matrix, and the vector X(x, t) is given by the formula

X(x, t, k) = −i�(k) − �(−i∂x)

k − (−i∂x)q(x, t). (8.5)

Equation (8.4) is equivalent to the statement that the N -vector differential 1-formW(x, t, k) defined by

W(x, t, k) = e−ikxI+ω(k)t [A(k)q(x, t) dx + A(k)X(x, t, k) dt], (8.6)

is a closed form.


Moreover, since we have diagonalised the system, we can uncouple Equa-tion (8.4) to N scalar equations of the same form. Hence Equation (8.4) is equiva-lent to the statement that the N differential 1-forms Wj(x, t, k), defined by

Wj(x, t, k)

= e−ikx+ωj (k)t [A(k)q(x, t) dx + A(k)X(x, t, k) dt]j , 1 � j � N, (8.7)

are closed (the subscript j denotes the j th component of the vector).It is straightforward to verify that if the matrix �(k) can be written in the

form (8.3) then Equation (8.2) can be written in the form (8.4), where X is givenby Equation (8.5). Indeed, Equation (8.4) is

ω(k)A(k)q(x, t) + A(k)qt (x, t) − (ikA(k) − A(k)∂x)X(x, t, k) = 0.

Replacing qt(x, t) by −�(−i∂x)q(x, t), we can write this equation as

ω(k)A(k)q(x, t) − A(k)�(−i∂x)q(x, t) −− iA(k)(k − (−i∂x))X(x, t, k) = 0.

Multiplying the latter by A(k)−1 and using Equation (8.3), we find

(�(k) − �(−i∂x))q(x, t) = i(k − (−i∂x))X(x, t, k).

Thus if X is defined by Equation (8.5), this equation is identically satisfied.Equation (8.4) is equivalent to the condition that W is a closed form. Indeed,

dW(x, t, k) = e−ikxI+ω(k)t [(A(k)qt (x, t) + A(k)∂x)X(x, t, k)) dt ∧ dx ++ (ω(k)A(k)q(x, t) − ikA(k)X(x, t, k)) dx ∧ dt] = 0,

where we have used the skew-symmetry of the wedge product.Assuming that the domain D in which the system is considered is simply

connected, the equation dW = 0 has two important consequences:(a) There exists a differential 0-form O(x, t, k) such that dO = W .Writing the N -vector O in the form

O = e−ikx+ω1(k)tM1· · ·

e−ikx+ωN(k)tMN

we find the N scalar equations

d[e−ikx+ωj (k)tMj ] = Wj(x, t, k), j = 1, . . . , N, (8.8)

where the scalars Wj are defined by (8.7). Note that the equation dO = W isequivalent to the statement that the following pair of ODEs, asssociated with the


function M(x, t, k), is compatible:{Mx − ikM = A(k)q(x, t), (8.9a)

Mt + ω(k)M = −A(k)X(x, t, k). (8.9b)

(b) The integral of the form W along the boundary of the simply connected Dvanishes:∮

∂D

W = 0. (8.10)

We call this equation the global relation.The spectral analysis of the form W means finding a vector solution M(x, t, k) =

(M1, . . . , MN)τ of (8.8) which is sectionally bounded with respect to the complexspectral variable k.

We now show how the knowledge of the function M yields an expression for thefunction q(x, t). Assume that the N × N matrix A(k) appearing in Equation (8.3)has the following large k behaviour:

A(k) = kA1 + A0 + O

(1

k

), k −→ ∞. (8.11)

Then Equation (8.9a) implies that

M = iA1q + iA0q + (A1q)x

k+ O

(1

k2

), k −→ ∞. (8.12)

To obtain a representation for the vector M , it is convenient to have a solution M

which decays as k → ∞. To obtain a decaying solution, we define the vector µ by

µ = M − iA1q.

Then the system (8.9) becomes{µx − ikµ = (A − kA1)q − i(A1q)x,

µt + ω(k)µ = −AX − iω(k)A1q − i(A1q)t .(8.13)

In this paper we consider only simple polygonal domains, namely 0 < t < T

and either 0 < x < ∞ or 0 < x < L. For such domains, which are alwayssimply connected, it is possible to find a solution µ of (8.13) which is a sectionallyholomorphic function of k. This means that there exists an oriented contour �

dividing the complex plane in a (+) region and a (−) region,� and that there existsa function µ which is holomorphic in each of these regions, µ = µ+ in the (+)

region and µ = µ− in the (−) region. Let the vector J (x, t, k) denote the ‘jump’between µ+ and µ−

µ+(x, t, k) − µ−(x, t, k) = J (x, t, k), k ∈ �.

� By convention, the (+) region lies to the left of the positive orientation.


An important feature of this approach is that J (x, t, k) has an explicit x and t

dependence. Indeed, since both µ+ and µ− satisfy Equations (8.13), it follows that

d[J e−ikxI+ω(k)t ] = 0.

Thus

µ+(x, t, k) − µ−(x, t, k) = eikxI−ω(k)tρ(k), k ∈ �, (8.14)

where the vector ρ(k) is a function depending only on k, called the spectral func-tion. If A(k) has the asymptotic behaviour given in (8.11), then

µ = iA0q + (A1q)x

k+ O

(1

k2

), k → ∞. (8.15)

Equations (8.14) and (8.15) define a Riemann–Hilbert problem for the sectionallyholomorphic function µ [1]. Its unique solution is

µ(x, t, k) = 1

2iπ

∫�

eilxI−ω(l)tρ(l)

l − kdl, k ∈ C. (8.16)

Comparing Equation (8.15) with the large k behaviour of Equation (8.16) it followsthat

iA0q + (A1q)x = 1

2π

∫�

eikxI−ω(k)tρ(k) dk. (8.17)

This equation provides an integral representation of q in terms of the spectral func-tion ρ(k). This function can be expressed as an integral of q and of its derivativeson the boundary of the domain. These boundary values are related by the globalrelation (8.10). The analysis of this relation can be used to identify all well posedboundary value problems and to obtain the spectral function ρ(k) in terms of thegiven data. For simple boundary conditions, ρ(k) can be computed in terms of thegiven data, using only algebraic manipulations of the global relation.

8.1.1. The Linearised Classical Boussinesq System

We now apply the general method outlined above to the particular case of thelinearised classical Boussinesq system.

Let the vector q = (η, u)τ satisfy the linearised version of the classical Boussi-nesq system (1.4). This system can be formally written in the form

ηt + ux = 0,

ut + (1 − ∂2x )

−1ηx = 0,(8.18)

thus

�(k) = 0 ik

ik

1 + k20

.


Equation (8.5) implies

X(x, t, k) = − i

k + i∂x

0 i(k + i∂x)

ik

1 + k2− ∂x

1 − ∂2x

0

q(x, t)

= 0 1

1 + ik∂x

(1 − ∂2x )(1 + k2)

0

q(x, t),

hence

X(x, t, k) = u

1

1 + k2[1 + (1 − ∂2

x )−1(∂2

x + ik∂x)]η

.

Using the second of Equations (8.18), i.e. (1 − ∂2x )

−1ηx = −ut , we find

X(x, t, k) = u

1

1 + k2(η − utx − ikut )

.

The matrix �(k) diagonalises to

ik√1 + k2

σ3, σ3 =(

1 00 −1

).

It is convenient to eliminate the square root by the uniformising change of variablek → k / 2 − 1 / 2k. After this change of variable, Equations (1.4) can be expressedin the form (8.4), with

ω(k) = ik − 1

k

k + 1k

σ3, A(k) =

2α(k)

k + 1k

α(k)

−2β(k)

k + 1k

β(k),

, (8.19)

q(x, t) =(

η

u

), X(x, t, k) =

u

4

(k + 1k)2

(η − utx − i

2

(k − 1

k

)ut)

.

We let

α = β = 1

2

(k + 1

k

).

Hence A(k) and X(x, t, k) are given by

A(k) =

11

2

(k + 1

k

)

−11

2

(k + 1

k

)

, X(x, t, k) =

u

4

(k + 1k)2

H(x, t, k)

,

(8.20)


with

H(x, t, k) = η − uxt − i

2

(k − 1

k

)ut . (8.21)

The formulation (8.4) is now explicitly determined. Equivalently, the Lax pairformulation (8.9) for this system involves the auxiliary 2-vector M = M(x, t, k),

Mx − i

2

(k − 1

k

)M = Aq,

Mt + ω(k)M = −AX.

More explicitly, for the two components of M = (M1, M2)τ , we have the two Lax

pairs

(M1)x − i

2

(k − 1

k

)M1 = η + 1

2

(k + 1

k

)u,

(M1)t + ik − 1

k

k + 1k

M1 = −u − 2

k + 1k

H,

(8.22)

and

(M2)x − i

2

(k − 1

k

)M2 = −η + 1

2

(k + 1

k

)u,

(M2)t − ik − 1

k

k + 1k

M2 = u − 2

k + 1k

H.

(8.23)

It is easy to verify that M2(x, t, −1/k) = M1(x, t, k). Hence it is sufficient toanalyse the equations satisfied by the scalar function M1. Since M1 ∼ iu(x, t),k → ∞, the function µ(x, t, k) defined by µ1 = M1 − iu(x, t) is of order O( 1

k)

as k → ∞. The Lax pair satisfied by µ is given by Equations (2.15). The spectralanalysis of this pair is implemented below.

8.2. PROOF OF PROPOSITION 5.2.

Let the functions q(x, t, k) and X(x, t, k) be given by (2.12b) and (2.13), respec-tively. Consider the (x, t)-plane, and for any fixed point (x0, t0), consider the lineintegral from (x0, t0) to (x, t). One can define the following ‘canonical’ solutions:

µj(x, t, k)

=∫ (x,t)

(xj ,tj )

ei2 (k− 1

k)(x−x′)−ω(k)(t−t ′)[q(x ′, t ′) dx ′ + X(x ′, t ′, k) dt ′], (8.24)

where (x1, t1) = (0, T ), (x2, t2) = (0, 0), (x3, t3) = (∞, t), and the paths ofintegration are shown in Figure 3.

By considering the boundedness properties of the exponential involved in thedefinition of these functions, it is not difficult to check that µ1(x, t, k), µ2(x, t, k)


and µ3(x, t, k) are analytic functions of k, which are bounded as k → ∞ in thefirst quadrant, the second quadrant and the lower half of the complex k-plane,respectively. This is a consequence of the following:

Rei

2

(k − 1

k

)= −k2

2

|k|2 + 1

|k|2 , Re(ω(k)) = −4k1k2

|1 + k2|2 ;

Rei

2

(k − 1

k

)∼ −k2

2|k|2 , k −→ 0,

Re(ω(k)) ∼ −k1k2, k −→ 0, = k1 + ik2.

Equations (8.24) imply

µi − µj = ei2 (k− 1

k)x−ω(k)tρij (k), i �= j.

ρij (k) =∫ zj

zi

e− i2 (k− 1

k)x+ω(k)t [q(x, t, k) dx + X(x, t, k) dt]. (8.25)

Note that the exponential appearing in the difference µi − µj decays as k → ∞,when k ∈ D+. Computing the integrals in (8.25) along paths parallel to the x and t

axes, see Figure 3, we find

µ1 − µ3 = ei2 (k− 1

k)x−ω(k)t ×

×{− 2

k + 1k

∫ T

0eω(k)s

((uxt − η) − 1

k(iut + u)

)(0, s) ds +

+∫ ∞

0e− i

2 (k− 1k)y

(η − iux + u

k

)(y, 0) dy

},

Figure 3. The solutions µ1, µ2 and µ3.


µ1 − µ2 = −ei2 (k− 1

k)x−ω(k)t 2

k + 1k

∫ T

0eiω(k)s ×

×(

(uxt − η) − 1

k(iut + u)

)(0, s) ds,

µ3 − µ2 = −ei2 (k− 1

k)x−ω(k)t

∫ ∞

0e− i

2 (k− 1k)y

(η − iux + u

k

)(y, 0) dy.

Let µ(x, t, k) be defined by µ = µ3 for k in C−, µ = µ1 in the first quadrant and

µ = µ2 in the second quadrant of the complex k-plane. This function is sectionallyanalytic, and it has the following asymptotic behaviour:

µ(x, t, k) = 2(η − iux)

ik+ O

(1

k2

), k −→ ∞,

(8.26)µ(x, t, 0) = −2iu(x, t) + O(k), k −→ 0.

Equation (8.25) together with the asymptotic behaviour of µ define a Riemann–Hilbert problem whose unique solution is given by

µ(x, t, k) = 1

2πip.v.

{∫ ∞

−∞e

i2 (l− 1

l)x−ω(l)t J1(l)

l − kdl +

+∫

∂D+e

i2 (l− 1

l)x−ω(l)t J2(l)

l − kdl

}, (8.27)

where

J1(k) =∫ ∞

0e− i

2 (k− 1k)y

(η(y, 0) − iux(y, 0) + u(y, 0)

k

)dy,

and

J2(k) = − 2

k + 1k

∫ T

0eω(k)s

(uxt (0, s)− η(0, s)− 1

k(iut (0, s) +u(0, s))

)ds.

Note that, using the definitions (5.1) and (5.2), we can write

J1(k) = η − iux + u

k, J2(k) = 2

k + 1k

f (k).

As k → 0, J1(k) ∼ u + k(η − iux) and J2(k) ∼ iut + u − k(uxt − η), the functionµ(x, t, 0) given by (8.27) is not well defined in a neighbourhood of l = 0 unlessthe integrals appearing in the representation (8.27) are interpreted as principal valueintegrals at k = 0,

p.v.

∫ ∞

−∞= lim

ε→0

(∫ −ε

−∞+

∫ ∞

ε

).


Indeed, the limits from above and below the real k axis of the relevant functions µj

yield the definition of principal value.The estimates (8.26) now yield

u(x, t) = 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

(η0 − iu′

0 + u0

k

)dk

k+

+∫

∂D+e

i2 (k− 1

k)x−ω(k)t 2f (k)

1 + k2dk

}, (8.28)

and

η(x, t) = iux(x, t) + 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

(η0 − iu′

0 + u0

k

)dk +

+∫

∂D+e

i2 (k− 1

k)x−ω(k)t 2kf (k)

1 + k2dk

}. (8.29)

Computing ux(x, t) from (8.28) and substituting the result into the formula (8.29),we obtain

η(x, t) = 1

4πp.v.

{∫ ∞

−∞e

i2 (k− 1

k)x−ω(k)t

(η0 − iu′

0 + u0

k

)1

2

(1 + 1

k2

)dk +

+∫

∂D+e

i2 (k− 1

k)x−ω(k)t f (k)

dk

k

}. (8.30)

Alternatively, using Equation (1.4a) to express η we obtain Equation (5.8).Equation (8.10) where ∂D is the boundary of the domain D = {(x, t) : 0 <

x < ∞, 0 < t < T } yields Equation (5.4).

References

1. Ablowitz, M. J. and Fokas, A. S.: Introduction and Applications of Complex Variables,Cambridge University Press, 2nd edn, 2003.

2. Amick, C. J.: Regularity and uniqueness of solutions to the Boussinesq system of equations,J. Differential Equations 54 (1984), 231–247.

3. Bona, J. L., Chen, M. and Saut, J. C.: Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media I, J. Nonlinear Sci. 12 (2002), 283–318.

4. Colin, T. and Ghidaglia, J. M.: An initial-boundary value problem for the Korteweg–deVriesequation posed on a finite interval, Adv. Diff. Eq. 6(12) (2001), 1463–1492.

5. Dougalis, V. A. and Pelloni, B.: Numerical modelling of two-way propagation of nonlineardispersive waves, Math. Comput. Simulation 55 (2001), 595–606.

6. Fokas, A. S.: A unified transform method for solving linear and certain nonlinear PDE’s, Proc.Roy. Soc. London Ser. A 453 (1997), 1411–1443.

7. Fokas, A. S.: On the integrability of linear and nonlinear PDEs, J. Math. Phys. 41 (2000), 4188.8. Fokas, A. S.: Two-dimensional linear PDE’s in a convex polygon, Proc. Roy. Soc. London Ser. A

457 (2001), 371–393.9. Fokas, A. S.: A new transform method for evolution PDEs, IMA J. Appl. Math. 67 (2002),

559–590.


10. Fokas, A. S.: Integrable nonlinear evolution equations on the half line, Comm. Math. Phys. 230(2002), 1–39.

11. Fokas, A. S. and Its, A. R.: The nonlinear Schrödinger equation on a finite domain, J. Phys. A,Math. Gen. 37 (2004), 6091–6114.

12. Fokas, A. S. and Pelloni, B.: Integral transforms, spectral representations and the d-barproblem, Proc. Roy. Soc. London Ser. A 456 (2000), 805–833.

13. Fokas, A. S. and Pelloni, B.: Two-point boundary value problems for linear evolution equations,Proc. Camb. Phil. Soc. 17 (2001), 919–935.

14. Fokas, A. S. and Pelloni, B.: A transform method for evolution PDEs on a finite interval,submitted to IMA J. Appl. Math. (in press).

15. Fokas, A. S. and Sung, L. Y.: Initial boundary value problems for linear evolution equations onthe half line, Ann. of Math. (in press).

16. Fokas, A. S. and Zyskin, M.: The fundamental differential form and boundary value problems,Quart. J. Mech. Appl. Math. 55 (2002), 457–479.

17. Pelloni, B.: Well-posed boundary value problems for linear evolution equations on a finiteinterval, Proc. Camb. Phil. Soc. 136 (2004), 361–382.

18. Schonbeck, M. E.: Existence of solutions for the Boussinesq system of equations, J. DifferentialEquations 42 (1981), 325–352.

19. Whitham, G. B.: Linear and Nonlinear Waves, Wiley, 1974.


Egorov’s Theorem for Transversally EllipticOperators on Foliated Manifolds andNoncommutative Geodesic Flow

YURI A. KORDYUKOVInstitute of Mathematics, Russian Academy of Sciences, Ufa, Russia. e-mail: [email protected]

(Received: 14 May 2002; in final form: 26 May 2004)

Abstract. The main result of the paper is Egorov’s theorem for transversally elliptic operators oncompact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flowin noncommutative geometry of Riemannian foliations.

Mathematics Subject Classifications (2000): 58J40, 58J42, 58B34.

Key words: noncommutative geometry, pseudodifferential operators, Riemannian foliations, geo-desic flow, transversally elliptic operators.

Introduction

Egorov’s theorem [8] is one of the fundamental results in microlocal analysis thatrelates the quantum evolution of pseudodifferential operators with the classicaldynamics of principal symbols.

Let P be a positive, self-adjoint, elliptic, first order pseudodifferential operatoron a compact manifold M with the positive principal symbol p ∈ S1(T ∗M \ 0).Let ft be the bicharacteristic flow of the operator P , that is, the Hamiltonian flowof p on T ∗M . For instance, one can consider P = √

�M , where �M is the Laplaceoperator of a Riemannian metric gM on M . Then the bicharacteristic flow of theoperator P is the geodesic flow of the metric gM .

Egorov’s theorem states that, for any pseudodifferential operator A of order 0with the principal symbol a ∈ S0(T ∗M \ 0), the operator A(t) = eitP Ae−itP is apseudodifferential operator of order 0. The principal symbol at ∈ S0(T ∗M \ 0) ofthis operator is given by the formula

at (x, ξ) = a(ft (x, ξ)), (x, ξ) ∈ T ∗M \ 0.

The main result of this paper is a version of Egorov’s theorem for transver-sally elliptic operators on compact foliated manifolds. This theorem is appliedto describe the noncommutative geodesic flow in noncommutative geometry ofRiemannian foliations.

98 YURI A. KORDYUKOV

1. Preliminaries and Main Results

1.1. TRANSVERSE PSEUDODIFFERENTIAL CALCULUS

Throughout in the paper, (M, F ) is a compact foliated manifold, E is a Hermitianvector bundle on M , dim M = n, dim F = p, p + q = n.

We will consider pseudodifferential operators, acting on half-densities. For anyvector bundle V on M , denote by |V |1/2 the associated half-density vector bundle.Let C∞(M, E) denote the space of smooth sections of the vector bundle E ⊗|T M|1/2, L2(M, E) the Hilbert space of square integrable sections of E⊗|T M|1/2,D ′(M, E) the space of distributional sections of E ⊗ |T M|1/2, D ′(M, E) =C∞(M, E)′, and Hs(M, E) the Sobolev space of order s of sections of E⊗|T M|1/2.Finally, let �m(M, E) denote the standard classes of pseudodifferential operators,acting in C∞(M, E).

We will use the classes �m,−∞(M, F , E) of transversal pseudodifferential op-erators. Let us briefly recall its definition, referring the reader to [14] for moredetails.

We will consider foliated coordinate charts �: U ⊂ M∼−→ I n on M with

coordinates (x, y) ∈ Ip × I q (I is the open interval (0, 1)) such that the restrictionof F to U is given by the sets y = const. We will always assume that foliatedcharts are regular. Recall that a foliated coordinate chart �: U ⊂ M

∼−→ I n

is called regular, if it admits an extension to a foliated coordinate chart �: V ⊂M

∼−→ (−2, 2)n with U ⊂ V .A map f : U ⊂ M → R

q is called a distinguished map, if f locally has theform prnq ◦ �, where �: V ⊂ U

∼−→ I n is a foliated chart and prnq: Rn = R

p ×R

q → Rq is the natural projection. Let Dx denote the set of germs of distinguished

maps from M to Rq at a point x ∈ M . For any leafwise continuous curve γ from

x to y, let hγ : Dx → Dy be the holonomy map associated with γ . This is thegeneralization of Poincaré’s first return map from flows to foliations.

Let �: U → Ip × I q, � ′: U ′ → Ip × I q , be two foliated charts, π = prnq ◦�: U → R

q , π ′ = prnq ◦� ′: U ′ → Rq the corresponding distinguished maps. The

foliation charts �, � ′ are called compatible, if, for any m ∈ U and m′ ∈ U ′ suchthat m = �−1(x, y), m′ = � ′−1

(x ′, y) with the same y, there is a leafwise path γ

from m to m′ such that the corresponding holonomy map hγ takes the germ πm ofthe map π at m to the germ π ′

m′ of the map π ′ at m′.Let �: U ⊂ M → Ip × I q, � ′: U ′ ⊂ M → Ip × I q , be two compatible

foliated charts on M equipped with trivializations of the vector bundle E overthem. Consider an operator A: C∞

c (U, E|U) → C∞c (U ′, E|U ′) given in the local

coordinates by the formula

Au(x, y) = (2π)−q

∫ei(y−y′)ηk(x, x ′, y, η)u(x ′, y ′) dx ′ dy ′ dη, (1)

where k ∈ Sm(Ip × Ip × I q × Rq, L(Cr )), u ∈ C∞

c (I n, Cr ), x ∈ Ip, y ∈ I q with

the Schwartz kernel, compactly supported in U × U ′ (here r = rank E).

EGOROV’S THEOREM AND NONCOMMUTATIVE GEODESIC FLOW 99

Recall that a function k ∈ C∞(Ip × Ip × I q × Rq, L(Cr )) belongs to the class

Sm(Ip × Ip × I q × Rq, L(Cr )), if, for any multiindices α and β, there exists a

constant Cαβ > 0 such that

|∂αη ∂

β

(x,x′,y)k(x, x ′, y, η)| � Cαβ(1 + |η|)m−|α|,

(x, x ′, y) ∈ Ip × Ip × I q, η ∈ Rq .

We will consider only classical symbols k, which can be represented as an asymp-totic sum k(x, x ′, y, η) ∼ ∑∞

j=0 θ(η)kz−j (x, x ′, y, η), where kz−j ∈ C∞(Ip×Ip×I q × (Rq\{0}), L(Cr )) is homogeneous in η of degree z − j , and θ is a smoothfunction on R

q such that θ(η) = 0 for |η| � 1, θ(η) = 1 for |η| � 2.The operator A extends to an operator in C∞(M, E) in a trivial way. The

resulting operator is said to be an elementary operator of class �m,−∞(M, F , E).The class �m,−∞(M, F , E) consists of all operators A in C∞(M, E), which

can be represented in the form A = ∑ki=1 Ai + K , where Ai are elementary oper-

ators of class �m,−∞(M, F , E), corresponding to some pairs �i, �′i of compatible

foliated charts, K ∈ �−∞(M, E). Put �∗,−∞(M, F , E) = ⋃m �m,−∞(M, F , E).

Let G be the holonomy groupoid of F . We will briefly recall its definition. Let∼h be the equivalence relation on the set of continuous leafwise paths γ : [0, 1] →M , setting γ1 ∼h γ2 if γ1 and γ2 have the same initial and final points and the sameholonomy maps. The holonomy groupoid G is the set of ∼h equivalence classesof continuous leafwise paths. G is equipped with the source and the range mapss, r: G → M defined by s(γ ) = γ (0) and r(γ ) = γ (1). We will identify a pointx ∈ M with the element of G given by the corresponding constant path: γ (t) =x, t ∈ [0, 1]. Recall also that, for any x ∈ M , the set Gx = {γ ∈ G: r(γ ) = x}is the covering of the leaf through the point x associated with the holonomy groupGx

x of this leaf, Gxx = {γ ∈ G : s(γ ) = x, r(γ ) = x}.

Any pair of compatible foliated charts �: U → Ip × I q, � ′: U ′ → Ip × I q

defines a foliated chart V → Ip × Ip × I q on G as follows. The coordinate patchV consists of all γ ∈ G from m = �−1(x, y) ∈ U to m′ = � ′−1

(x ′, y) ∈ U ′ suchthat the corresponding holonomy map hγ takes the germ πm of the distinguishedmap π = prnq ◦ � at m to the germ π ′

m′ of the distinguished map π ′ = prnq ◦ � ′ atm′, and the coordinate map takes such a γ to (x, x ′, y) ∈ Ip × Ip × I q .

Denote by N∗F the conormal bundle to F . For any γ ∈ G, s(γ ) = x, r(γ ) =y, the codifferential of the corresponding holonomy map defines a linear mapdh∗

γ : N∗y F → N∗

x F . Let FN be the linearized foliation in N∗F = N∗F \ 0

(cf., for instance, [20]). The leaf of the foliation FN through ν ∈ N∗F is the set ofall points dh∗

γ (ν) ∈ N∗F , where γ ∈ G, r(γ ) = π(ν) (here π : T ∗M → M is thebundle map). The leaves of the foliation FN have trivial holonomy. Therefore, theholonomy groupoid GFN

of FN consists of all pairs (γ, ν) ∈ G × N∗F such thatr(γ ) = π(ν) with the source map sN : GFN

→ N∗F , sN(γ, ν) = dh∗γ (ν) and the

range map rN : GFN→ N∗F , rN(γ, ν) = ν. We have a map πG: GFN

→ G givenby πG(γ, ν) = γ .


Denote by π∗E the lift of the vector bundle E to N∗F via the bundle mapπ : N∗F → M and by L(π∗E) the vector bundle on GFN

, whose fiber at apoint (γ, ν) ∈ GFN

is the space L((π∗E)sN (γ,ν), (π∗E)rN (γ,ν)) of linear maps from

(π∗E)sN (γ,ν) to (π∗E)rN(γ,ν). There is a natural foliation GN on GFN. The leaf of

GN through a point (γ, ν) ∈ GFNis the set of all (γ ′, ν ′) ∈ GFN

such that ν and ν ′lie in the same leaf in FN . Let |T GN |1/2 be the line bundle of leafwise half-densitieson GFN

with respect to the foliation GN . It is easy to see that

|T GN |1/2 = r∗N(|T FN |1/2) ⊗ s∗

N(|T FN |1/2),

where s∗N(|T FN |1/2) and r∗

N(|T FN |1/2) denote the lifts of the line bundle |T FN |1/2

of leafwise half-densities on N∗F via the source and the range mappings sN andrN , respectively.

A section k ∈ C∞(GFN, L(π∗E) ⊗ |T GN |1/2) is said to be properly supported,

if the restriction of the map r: GFN→ N∗F to supp k is a proper map. Con-

sider the space C∞prop(GFN

, L(π∗E) ⊗ |T GN |1/2) of smooth, properly supportedsections of L(π∗E) ⊗ |T GN |1/2. One can introduce the structure of involutivealgebra on C∞

prop(GFN, L(π∗E) ⊗ |T GN |1/2) by the standard formulas (cf. (8)).

Let Sm(GFN, L(π∗E) ⊗ |T GN |1/2) be the space of all s ∈ C∞

prop(GFN, L(π∗E) ⊗

|T GN |1/2) homogeneous of degree m with respect to the action of R given by themultiplication in the fibers of the vector bundle πG: GFN

→ G. By [14], there isthe half-density principal symbol mapping

σ : �m,−∞(M, F , E) −→ Sm(GFN, L(π∗E) ⊗ |T GN |1/2), (2)

which satisfies

σm1+m2(AB) = σm1(A)σm2(B), σm1(A∗) = σm1(A)∗

for any A ∈ �m1,−∞(M, F , E) and B ∈ �m2,−∞(M, F , E).

EXAMPLE 1.1. Consider a foliated coordinate chart �: U ⊂ M∼−→ I n on M

with coordinates (x, y) ∈ Ip × I q . One has the corresponding coordinate chartin T ∗M with coordinates given by (x, y, ξ, η) ∈ Ip × I q × R

p × Rq . In these

coordinates, the restriction of the conormal bundle N∗F to U is given by theequation ξ = 0. So we have a coordinate chart �n: U1 ⊂ N∗F ∼−→ Ip × I q × R

q

on N∗F with the coordinates (x, y, η) ∈ Ip × I q × Rq . The coordinate chart �n

is a foliated coordinate chart for the linearized foliation FN , and the restriction ofFN to U1 is given by the level sets y = const, η = const.

Now let �: U ⊂ M → Ip × I q, � ′: U ′ ⊂ M → Ip × I q , be two com-patible foliated charts on M . Then the corresponding foliated charts �n: U1 ⊂N∗F → Ip × I q × R

q, � ′n: U ′

1 ⊂ N∗F → Ip × I q × Rq, are compatible

with respect to the foliation FN . So they define a foliated chart V on the foliatedmanifold (GFN

, GN) with the coordinates (x, x ′, y, η) ∈ Ip × Ip × I q × Rq , and

the restriction of GN to V is given by the level sets y = const, η = const. The


principal symbol σm(A) of an operator A given by the formula (1) is the half-density km(x, x ′, y, η) |dx|1/2 |dx ′|1/2, where km is the top degree homogeneouscomponent of k. It can be checked that this half-density is globally defined as anelement of the space Sm(GFN

, L(π∗E) ⊗ |T GN |1/2).

1.2. TRANSVERSE BICHARACTERISTIC FLOW

For any operator P ∈ �m(M, E), let σP denote the transversal principal symbolof P , which is the restriction of its principal symbol to N∗F . We say that P istransversally elliptic, if σP (ν) is invertible for any ν ∈ N∗F .

Consider a transversally elliptic operator A ∈ �2(M, E) which has the scalarprincipal symbol and the holonomy invariant transverse principal symbol. Here theholonomy invariance of the transversal principal symbol σA ∈ C∞(N∗F ) meansthat it is constant along the leaves of the foliation FN :

σA(dh∗γ (ν)) = σA(ν), γ ∈ G, ν ∈ N∗

r(γ )F .

Let a2 ∈ S2(T ∗M) be the principal symbol of A. (Here T ∗M = T ∗M \ 0.)Take any scalar elliptic symbol p ∈ S1(T ∗M), which is equal to

√a2 in some

conic neighborhood of N∗F . Denote by Xp the Hamiltonian vector field of p onT ∗M . Since N∗F is a coisotropic submanifold in T ∗M and T FN is the symplecticorthogonal complement of T (N∗F ), one can show that Xp is tangent to N∗F , andits restriction to N∗F (denoted also by Xp) is an infinitesimal transformation of thefoliation FN , i.e. for any vector field X on N∗F , tangent to FN , the commutator[Xp, X] is tangent to FN . It follows that the Hamiltonian flow ft of p preservesN∗F , and its restriction to N∗F (denoted by ft ) preserves the foliation FN , thatis, takes any leaf of FN to a leaf.

Let τ = T N∗F /T FN be the normal space to the foliation FN and πtr:T N∗F → τ the natural projection. For any (γ, ν) ∈ GFN

, let dH(γ,ν): τdh∗γ (ν) →

τν be the corresponding linear holonomy map. The differential of the map (sN, rN):GFN

→ N∗F × N∗F at a point (γ, ν) ∈ GFNdefines an inclusion of T(γ,ν)GFN

into Tdh∗γ (ν)N

∗F × TνN∗F , and its image consists of all (X, Y ) ∈ Tdh∗

γ (ν)N∗F ×

TνN∗F such that

πtr(Y ) = dH(γ,ν)(πtr(X)). (3)

Since Xp is an infinitesimal transformation of the foliation FN , one can see that,for any (γ, ν) ∈ GFN

, the pair (Xp(dh∗γ (ν)), Xp(ν)) ∈ Tdh∗

γ (ν)N∗F × TνN

∗Fsatisfies (3). Therefore, there exists a unique vector field Hp on GFN

such thatdsN(Hp) = Xp and drN(Hp) = Xp. Let Ft be the flow on GFN

determined by thevector field Hp. It is easy to see that sN ◦ Ft = ft ◦ sN , rN ◦ Ft = ft ◦ rN and theflow Ft preserves the foliation GN .


DEFINITION 1.2. Let P = √A be an (unbounded) linear operator in L2(M, E),

where A ∈ �2(M, E) is an essentially self-adjoint, transversally elliptic operator,which has the scalar principal symbol and the holonomy invariant transverse prin-cipal symbol. The transversal bicharacteristic flow of P is the one-parameter groupF ∗

t of automorphisms of the involutive algebra C∞prop(GFN

, |T GN |1/2) induced bythe flow Ft on GFN

.

Remark 1.3. It is easy to see that the definition of transversal bicharacteristicflow is independent of a choice of the elliptic extension p.

EXAMPLE 1.4. Consider a foliated coordinate chart �: U ⊂ M∼−→ I n on M

with coordinates (x, y) ∈ Ip × I q . Let p be a positive, smooth homogeneous ofdegree 1 function on I n×(Rn\{0}) (a scalar elliptic principal symbol) such that thecorresponding transversal principal symbol σP is holonomy invariant. This means

p(x, y, 0, η) = p(y, η), x ∈ Ip, y ∈ I q, η ∈ Rq

with some function p. The Hamiltonian vector field Xp on I n × Rn is given by

Xp = ∂p

∂ξ

∂

∂x− ∂p

∂x

∂

∂ξ+ ∂p

∂η

∂

∂y− ∂p

∂y

∂

∂η,

and its restriction to N∗F |U ∼= Ip × I q × Rq is given by

Xp(x, y, η) = ∂p

∂ξ(x, y, 0, η)

∂

∂x+ ∂p

∂η(y, η)

∂

∂y− ∂p

∂y(y, η)

∂

∂η,

(x, y, η) ∈ Ip × I q × Rq .

The fact that Xp is an infinitesimal transformation of the foliation FN means thatits transverse part

∂p

∂η(y, η)

∂

∂y− ∂p

∂y(y, η)

∂

∂η

is independent of x. The corresponding vector field Hp on GFNis given by

Hp(x, x ′, y, η)

= ∂p

∂ξ(x, y, 0, η)

∂

∂x+ ∂p

∂ξ(x ′, y, 0, η)

∂

∂x ′ + ∂p

∂η(y, η)

∂

∂y− ∂p

∂y(y, η)

∂

∂η,

(x, x ′, y, η) ∈ Ip × Ip × I q × Rq .

Finally, the transversal bicharacteristic flow is given by the action of the flowFt determined by the vector field Hp on the space of half-densities of the formkm(x, x ′, y, η) |dx|1/2 |dx ′|1/2.


Remark 1.5. The construction of the transversal bicharacteristic flow providesan example of what can be called noncommutative symplectic (or, maybe, bet-ter, Poisson) reduction. Here symplectic reduction means the following procedure[16, Chapter III, Section 14] (see also [17, 18]).

Let (X, ω) be a symplectic manifold, and Y a submanifold of X such that the2-form ωY induced by ω on Y is of constant rank. Let FY be the characteristic folia-tion of Y relative to ωY . If the foliation FY is simple, that is, it is given by the fibersof a surjective submersion p of Y to a smooth manifold B, then B has a uniquesymplectic form ωB such that p∗ωB = ωY . The symplectic manifold (B, ωB) issaid to be the reduced symplectic manifold associated with Y . In a particular casewhen the submanifold Y is the preimage of a point under the momentum mapassociated with the Hamiltonian action of a Lie group, the symplectic reductionassociated with Y is the Mardsen–Weinstein symplectic reduction [19].

Moreover (see, for instance, [16, Chapter III, Theorem 14.6]), if Y is invariantunder the Hamiltonian flow of a Hamiltonian H ∈ C∞(X) (this is equivalentto the fact that (dH)|Y is constant along the leaves of the characteristic foliationFY ), there exists a unique function H ∈ C∞(B), called the reduced Hamiltonian,such that H |Y = H ◦ p. Furthermore, the map p projects the restriction of theHamiltonian flow of H to Y to the reduced Hamiltonian flow on B defined by thereduced Hamiltonian H .

Now let (M, F ) be a smooth foliated manifold. Consider the symplectic re-duction associated with the coisotropic submanifold Y = N∗F in the symplecticmanifold X = T ∗M . The corresponding characteristic foliation FY is the lin-earized foliation FN . In general, the leaf space N∗F /FN is not a smooth manifold.Following ideas of the noncommutative geometry in the sense of A. Connes, onecan treat the algebra C∞

prop(GFN, |T GN |1/2) as a noncommutative analogue of an

algebra of smooth functions on N∗F /FN . The symplectic reduction procedure isapplied to the Hamiltonian flow ft of a function p satisfying the assumptions givenin the beginning of this section, yielding the transversal bicharacteristic flow F ∗

t asthe corresponding reduced Hamiltonian flow on N∗F /FN . Following the ideas of[2, 28], one can interpret the algebra C∞

prop(GFN, |T GN |1/2) as a noncommutative

Poisson manifold and the flow F ∗t as a noncommutative Hamiltonian flow.

EXAMPLE 1.6. Let (M, F ) be a compact Riemannian foliated manifoldequipped with a bundle-like metric gM . Let F = T F be the tangent bundle toF , H the orthogonal complement to F , and gH the restriction of gM to H . Bydefinition, a Riemannian metric gM on M is called bundle-like, if it satisfies one ofthe following equivalent conditions (see, for instance, [20, 23]):

(1) For any continuous leafwise path γ from x to y, the corresponding linearholonomy map dhγ : TxM/TxF → TyM/TyF is an isometry with respectto the Riemannian structures on TxM/TxF and TyM/TyF induced by themetric gM ;


(2) If gH is written as gH = ∑αβ gαβ(x, y)θαθβ in some foliated chart with coor-

dinates (x, y) ∈ Ip × I q , where θα ∈ H ∗ is the (unique) lift of dyα under theprojection Ip × I q → I q , then gαβ is independent of x, gαβ(x, y) = gαβ(y).

The decomposition F ⊕ H = T M induces a bigrading on∧

T ∗M:

∧k

T ∗M =k⊕

i=0

i,k−i∧T ∗M,

where∧i,j

T ∗M = ∧iF ∗ ⊗ ∧j

H ∗. In this bigrading, the de Rham differential dcan be written as

d = dF + dH + θ,

where dF and dH are first-order differential operators (the tangential de Rhamdifferential and the transversal de Rham differential accordingly), and θ is a zeroorder differential operator.

The transverse signature operator is a first order differential operator inC∞(M,

∧H ∗) given by

DH = dH + d∗H ,

and the transversal Laplacian is a second order transversally elliptic differentialoperator in C∞(M,

∧H ∗) given by

�H = D2H .

The principal symbol σ(�H) of �H is given by

σ(�H)(x, ξ) = gH (ξ, ξ)Ix, (x, ξ) ∈ T ∗M,

and holonomy invariance of the transversal principal symbol is equivalent to theassumption on the metric gM to be bundle-like.

Take any function p2 ∈ C∞(T ∗M), which coincides with√

σ(�H) in someconical neighborhood of N∗F . The restriction of the Hamiltonian flow of p2 toN∗F coincides with the restriction Gt of the geodesic flow gt of the Riemannianmetric gM to N∗F , which is the transversal bicharacteristic flow of the operator〈DH 〉 = √

�H + I .Finally, if F is given by the fibers of a Riemannian submersion f : M → B,

then there is a natural isomorphism N∗mF → T ∗

f (m)B, and, under this isomorphism,the transversal geodesic flow Gt on N∗F corresponds to the geodesic flow T ∗B(see, for instance, [21, 23]).


1.3. EGOROV’S THEOREM

Let D ∈ �1(M, E) be a formally self-adjoint, transversally elliptic operator suchthat D2 has the scalar principal symbol and the holonomy invariant transverse prin-cipal symbol. By [14], the operator D is essentially self-adjoint with initial domainC∞(M, E). Define an unbounded linear operator 〈D〉 in the space L2(M, E) as

〈D〉 = (D2 + I )1/2.

By the spectral theorem, the operator 〈D〉 is well-defined as a positive, self-adjointoperator in L2(M, E). The operator 〈D〉2 ∈ �2(M, E) is a bounded operator fromH 2(M, E) to L2(M, E). Hence, by interpolation, 〈D〉 defines a bounded operatorfrom H 1(M, E) to L2(M, E) and H 1(M, E) is contained in the domain of 〈D〉 inL2(M, E).

By the spectral theorem, the operator 〈D〉s = (D2 + I )s/2 is a well-definedpositive self-adjoint operator in H = L2(M, E) for any s ∈ R, which is unboundedif s > 0. For any s � 0, denote by H s the domain of 〈D〉s , and, for s < 0,H s = (H−s)∗. Put also H∞ = ⋂

s�0 H s, H−∞ = (H∞)∗. It is clear thatHs(M, E) ⊂ H s for any s � 0 and H s ⊂ Hs(M, E) for any s < 0. In particular,C∞(M, E) ⊂ H s for any s.

We say that a bounded operator A in H∞ belongs to L(H−∞, H∞) (resp.K(H−∞, H∞)), if, for any s and r , it extends to a bounded (resp. compact) opera-tor from H s to H r , or, equivalently, the operator 〈D〉rA〈D〉−s extends to a bounded(resp. compact) operator in L2(M, E). It is easy to see that L(H−∞, H∞) is ainvolutive subalgebra in L(H) and K(H−∞, H∞) is its ideal. We also introducethe class L1(H−∞, H∞), which consists of all operators from K(H−∞, H∞)

such that, for any s and r , the operator 〈D〉rA〈D〉−s is a trace class operator inL2(M, E). It should be noted that any operator K with the smooth kernel belongsto L1(H−∞, H∞).

As an operator acting on half-densities, any operator P ∈ �m(M) has the sub-principal symbol which is the well-defined homogeneous of degree m − 1 smoothfunction on T ∗M \ 0 given in local coordinates by the formula

psub = pm−1 − 1

2i

n∑j=1

∂2pm

∂xj∂ξj

, (4)

where pm−1 and pm are the homogeneous components of the complete symbolof P of degree m − 1 and m respectively. Observe that psub = 0 if P is a real,self-adjoint, differential operator of even order. In particular, this holds for thetransversal Laplacian �H on functions.

By the spectral theorem, the operator 〈D〉 defines a strongly continuous groupeit〈D〉 of bounded operators in L2(M, E). Consider a one-parameter group �t of∗-automorphisms of the algebra L(L2(M, E)) defined by

�t(T ) = eit〈D〉T e−it〈D〉, T ∈ L(L2(M, E)).

The main result of the paper is the following theorem.


THEOREM 1.7. Let D ∈ �1(M, E) be a formally self-adjoint, transversallyelliptic operator such that D2 has the scalar principal symbol and the holonomyinvariant transverse principal symbol.

(1) For any K ∈ �m,−∞(M, F , E), there exists an operator K(t) ∈ �m,−∞(M,

F , E) such that �t(K)−K(t), t ∈ R, is a smooth family of operators of classL1(H−∞, H∞).

(2) If, in addition, E is the trivial line bundle, and the subprincipal symbol ofD2 is zero, then, for any K ∈ �m,−∞(M, F ) with the principal symbol k ∈Sm(GFN

, |T GN |1/2), the principal symbol k(t) ∈ Sm(GFN, |T GN |1/2) of the

operator K(t) is given by k(t) = F ∗t (k), where F ∗

t is the transverse bicharac-teristic flow of the operator 〈D〉.

Remark 1.8. Theorem 1.7 implies Egorov’s theorem for elliptic operators oncompact Riemannian orbifolds. An m-dimensional orbifold M is a Hausdorff, sec-ond countable topological space, which is locally diffeomorphic to the quotientof R

m by a finite group of diffeomorphisms �. The notion of orbifold was firstintroduced by Satake in [24], where a different name, V -manifold, was used. Werefer the reader to [24, 12, 3] for expositions of orbifold theory. It is well-known(see, for instance, [13]) that any orbifold M is diffeomorphic to the orbifold ofG orbits of an action of a compact Lie group G on a compact manifold P wherethe action has finite isotropy groups (actually, one can take P to be the orthogonalframe bundle of M and G = O(m)). The orbits of this action are the leaves of afoliation F on P . We will use a natural isomorphism of the space C∞(M) with thespace C∞(P )G of G invariant functions on P . A pseudodifferential operator A inC∞(P ) can be defined as an operator acting on C∞(P )G which is the restriction ofa G equivariant pseudodifferential operator A in C∞(M). The operator A is ellipticiff the corresponding operator A is transversally elliptic with respect to the foliationF . The orthogonal projection � on the space of G-invariant functions in C∞(P )

is a transversal pseudodifferential operator of class �0,−∞(P, F ). It follows thata pseudodifferential operator A in C∞(M) coincides with the restriction of theoperator �A� ∈ �0,−∞(P, F ) to C∞(P )G.

Fix Riemannian metrics gM on M and gP on P such that the quotient mapP → M is a Riemannian submersion. So gP is a bundle-like metric on the foliatedmanifold (P, F ). One can show that the associated transverse Laplacian �H isG-invariant and the Laplacian �M on M coincides with the restriction of �H toC∞(P )G. Therefore, we have

eit (�M+I )1/2Ae−it (�M+I )1/2 = �eit (�H +I )1/2

Ae−it (�H +I )1/2�

= �eit (�H +I )1/2(�A�)e−it (�H +I )1/2

�.

By Theorem 1.7, it follows that the operator eit (�M+I )1/2Ae−it (�M+I )1/2

is a pseudo-differential operator on M and one can describe its principal symbol as in theclassical Egorov’s theorem. The details will be given elsewhere.


1.4. NONCOMMUTATIVE GEODESIC FLOW ON FOLIATED MANIFOLDS

As stated in [14], any operator D, satisfying the assumptions of Section 1.3, de-fines a spectral triple in the sense of Connes’ noncommutative geometry. In thissetting, Theorem 1.7 has a natural interpretation in terms of the correspondingnoncommutative geodesic flow. First, we recall general definitions [6, 5].

Let (A, H , D) be a spectral triple [5]. Here

(1) A is an involutive algebra;(2) H is a Hilbert space equipped with a ∗-representation of the algebra A (we

will identify an element a ∈ A with the corresponding operator in H );(3) D is an (unbounded) self-adjoint operator in H such that

(a) for any a ∈ A, the operator a(D − i)−1 is a compact operator in H ;(b) D almost commutes with any a ∈ A in the sense that [D, a] is bounded

in H .

As above, let 〈D〉 = (D2 + I )1/2. By δ, we denote the (unbounded) derivativeon L(H) given by

δ(T ) = [〈D〉, T ], T ∈ Dom δ ⊂ L(H). (5)

Let OPα be the space of operators in H of order α, that means that P ∈ OPα

iff P 〈D〉−α ∈ ⋂n Dom δn. In particular, OP0 = ⋂

n Dom δn. Denote by OP00

the space of all operators P ∈ OP0 such that 〈D〉−1P and P 〈D〉−1 are compactoperators in H . We also say that P ∈ OPα

0 if P 〈D〉−α and 〈D〉−αP are in OP00. It

is easy to see that OP−∞0 = ⋂

α OPα0 coincides with K(H−∞, H∞).

We will assume that (A, H , D) is smooth. This means that, for any a ∈ A,the bounded operators a and [D, a] in H belong to OP0. Let B be the algebra ofbounded operators in H generated by the set of all operators of the form δn(a) witha ∈ A and n ∈ N. Furthermore, we assume that the algebra B is contained in OP0

0.In particular, this implies that (B, H , D) is a spectral triple in the above sense.

In [6, 5], the definition of the algebra �∗(A) of pseudodifferential operatorswas given for a unital algebra A. In the case under consideration, the algebra Ais non-unital, that, roughly speaking, means that the associated geometric space isnoncompact. Therefore, we must take into account behavior of pseudodifferentialoperators at “infinity”. Next we define an algebra �∗

0 (A), which can consideredas an analogue of the algebra of pseudodifferential operators on a noncompactRiemannian manifold, whose symbols and all its derivatives of any order vanish atinfinity. In particular, the assumptions on the spectral triple made above mean thatthe algebra A consists of smooth “functions”, vanishing at “infinity” with all itsderivatives of any order.

Define �∗0 (A) as the set of (unbounded) operators in H , which admit an as-

ymptotic expansion:

P ∼+∞∑j=0

bq−j 〈D〉q−j , bq−j ∈ B, (6)


that means that, for any N ,

P − (bq〈D〉q + bq−1〈D〉q−1 + · · · + b−N 〈D〉−N) ∈ OP−N−10 .

By an easy modification of the proof of Theorem B.1 in [6, Appendix B], one canprove that �∗

0 (A) is an algebra. Let C0 be the algebra C0 = OP00 ∩ �∗

0 (A), and C0

the closure of C0 in L(H).For any T ∈ L(H), define

αt(T ) = eit〈D〉T e−it〈D〉, t ∈ R. (7)

As usual, K denotes the ideal of compact operators in H . The following definitionsare motivated by the work of Connes [5].

DEFINITION 1.9. Under the current assumptions on a spectral triple (A, H , D),the unitary cotangent bundle S∗A is defined as the quotient of the C∗-algebragenerated by all αt(C0), t ∈ R and K by K .

DEFINITION 1.10. Under the current assumptions on a spectral triple (A, H , D),the noncommutative geodesic flow is the one-parameter group αt of automorphismsof the algebra S∗A defined by (7).

We consider spectral triples (A, H , D) associated with a compact foliated Rie-mannian manifold (M, F ) [14]:

(1) The involutive algebra A is the algebra C∞c (G, |T G|1/2);

(2) The Hilbert space H is the space L2(M, E) of L2-sections of a holonomyequivariant Hermitian vector bundle E, on which an element k of the algebraA is represented via the ∗-representation RE (see below for a definition);

(3) The operator D is a first order self-adjoint transversally elliptic operator withthe holonomy invariant transversal principal symbol such that the operator D2

has the scalar principal symbol.

We recall briefly the definitions of the structure of involutive algebra on A andof the representation RE . Let α ∈ C∞(M, |T F |1/2) be a strictly positive, smooth,leafwise half-density. One can lift α to a strictly positive, leafwise half-densityνx = s∗α ∈ C∞(Gx, |T Gx |1/2) via the covering map s: Gx → Lx (Lx is the leafthrough a point x ∈ M). In the presence of ν, the space A = C∞

c (G, |T G|1/2) isnaturally identified with C∞

c (G). We also assume, for simplicity, that there exists aholonomy invariant, smooth, transverse half-density � ∈ C∞(M, |T M/T F |1/2).Recall that the holonomy invariance of � means that dh∗

γ (�(y)) = �(x)

for any γ ∈ G, s(γ ) = x, r(γ ) = y, where the map dh∗γ : |TyM/TyF |1/2 →

|TxM/TxF |1/2 is induced by the corresponding linear holonomy map.The multiplication and the involution in A are given by the formulas

(k1 ∗ k2)(γ ) =∫

Gx

k1(γ′−1γ )k2(γ

′) dνx(γ ′), γ ∈ Gx,

k∗(γ ) = k(γ −1), γ ∈ G, (8)


where k, k1, k2 ∈ A.An Hermitian vector bundle E on M is holonomy equivariant, if it is equipped

with an isometric action

T (γ ): Ex −→ Ey, γ ∈ G, γ : x −→ y

of G in fibers of E. Using the fixed half-densities α and �, one can identify ele-ments of L2(M, E) with square integrable sections of the bundle E. Then, for anyu ∈ L2(M, E), the section RE(k)u ∈ L2(M, E) is defined by the formula

RE(k)u(x) =∫

Gx

k(γ ) T (γ )[u(s(γ ))] dνx(γ ), x ∈ M.

It was stated in [14] that the spectral triple (A, H , D) associated with a compactfoliated Riemannian manifold is smooth. Recall that this means that, for any a ∈ A,a and [D, a] belong to OP0 = ⋂

n Dom δn. There is a gap in the proof of this factgiven in [14]. In this paper, we give a correct proof (cf. Theorem 3.2 below). InTheorem 3.2, we also prove that, in the case in question, the algebra B mentionedabove is contained in OP0

0.For any ν ∈ N∗F , there is a natural ∗-representation Rν of the algebra S0(GFN

,

|T GN |1/2) in L2(GνFN

, s∗N(π∗E)). For its definition, we will use the strictly positive,

leafwise half-density µν ∈ C∞(GνFN

, |T GN |1/2) induced by α and the correspond-ing isomorphism S0(GFN

, |T GN |1/2) ∼= S0(GFN). Since E is a holonomy equivari-

ant vector bundle, the bundle π∗E is also holonomy equivariant. The action of GFN

in fibers of π∗E,

π∗T (γ, ν): (π∗E)dh∗γ (ν) −→ (π∗E)ν, (γ, ν) ∈ GFN

,

is given by the formula π∗T (γ, ν) = T (γ ), where we use the natural isomor-phisms (π∗E)dh∗

γ (ν) = Ex and (π∗E)ν = Ey . For any k ∈ S0(GFN) and u ∈

L2(GνFN

, s∗N(π∗E)), the section Rν(k)u ∈ L2(Gν

FN, s∗

N(π∗E)) is given by theformula

Rν(k)u(γ, ν) =∫

GνFN

k((γ ′, ν)−1(γ, ν))π∗T (γ ′, ν)[u(γ ′, ν)] dµν(γ ′, ν),

(γ, ν) ∈ GνFN

.

It follows from the direct integral decomposition

L2(GFN, s∗

N(π∗E)) =∫

N∗FL2(Gν

FN, s∗

N(π∗E)) dν,

that, for any k ∈ S0(GFN, |T GN |1/2), the continuous family {Rν(k) ∈ L(L2(Gν

FN,

s∗N(π∗E))) : ν ∈ N∗F } defines a bounded operator in L2(GFN

, s∗N(π∗E)). We

will identify k ∈ S0(GFN, |T GN |1/2) with the corresponding bounded operator

in L2(GFN, s∗

N(π∗E)) and denote by S0(GFN, |T GN |1/2) the closure of S0(GFN

,


|T GN |1/2) in the uniform operator topology of L(L2(GFN, s∗

N(π∗E))). The transver-sal bicharacteristic flow F ∗

t of the operator 〈D〉 extends by continuity to a stronglycontinuous one-parameter group of automorphisms of S0(GFN

, |T GN |1/2).The following theorem gives a description of the associated noncommutative

geodesic flow in the scalar case.

THEOREM 1.11. Let (A, H , D) be a spectral triple associated with a compactfoliated Riemannian manifold (M, F ) as above with E, being the trivial holonomyequivariant line bundle. Assume that the subprincipal symbol of D2 vanishes. Thereexists a surjective homomorphism of involutive algebras P : S∗A → S0(GFN

,

|T GN |1/2) such that the following diagram commutes:

S∗Aαt

P

S∗A

P

S0(GFN, |T GN |1/2)

F ∗t

S0(GFN, |T GN |1/2).

(9)

2. Proof of the Main Theorem

2.1. THE CASE OF ELLIPTIC OPERATOR

Let (M, F ) be a compact foliated manifold, E a Hermitian vector bundle on M .In this section, we will assume that D ∈ �1(M, E) is a formally self-adjoint,elliptic operator such that D2 has the scalar principal symbol and the holonomyinvariant transverse principal symbol. Then P = 〈D〉 ∈ �1(M, E) is a self-adjointelliptic operator with the positive, scalar principal symbol p and the holonomyinvariant transversal principal symbol. In this case, the elliptic extension p of p

introduced in Section 1.2 can be taken to be equal to p, p = p. Therefore, if wedenote by Xp the Hamiltonian vector field of p on T ∗M , then the vector field Hp

can be described as a unique vector field on GFNsuch that dsN(Hp) = Xp and

drN(Hp) = Xp. Similarly, one can define the transverse bicharacteristic flow F ∗t

of P as in Definition 1.2, using p instead of p. The following theorem is slightlystronger than Theorem 1.7.

THEOREM 2.1. For any K ∈ �0,−∞(M, F , E), the operator �t(K) =eitP Ke−itP is an operator of class �0,−∞(M, F , E).

If E is the trivial line bundle, and the subprincipal symbol of D2 vanishes, then,for any K ∈ �0,−∞(M, F ) with the principal symbol k ∈ S0(GFN

, |T GN |1/2),the operator �t(K) has the principal symbol k(t) ∈ S0(GFN

, |T GN |1/2) given byk(t) = F ∗

t (k).


Proof. For the proof, we use theory of Fourier integral operators (see, for in-stance, [11, 26, 27]). Recall that a Fourier integral operator on M is a linear operatorF : C∞(M) → D ′(M), represented microlocally in the form

Fu(x) =∫

eφ(x,y,θ)a(x, y, θ) u(y) dy dθ, (10)

where x ∈ X ⊂ Rn, y ∈ Y ⊂ R

n, θ ∈ RN \ 0. Here a(x, y, θ) ∈ Sm(X × Y × R

N)

is an amplitude, φ is a nondegenerate phase function.Consider the smooth map from X × Y × R

N to T ∗X × T ∗Y given by

(x, y, θ) �−→ (x, φx(x, y, θ), y, −φy(x, y, θ)).

The image of the set

�φ = {(x, y, θ) ∈ X × Y × RN : φθ(x, y, θ) = 0}

under this map turns out to be a homogeneous canonical relation �φ in T ∗X×T ∗Y .(Recall that a closed conic submanifold C ∈ T ∗(X×Y )\0 is called a homogeneouscanonical relation, if it is Lagrangian with respect to the 2-form ωX − ωY , whereωX, ωY are the canonical symplectic forms in T ∗X, T ∗Y accordingly.)

The Fourier integral operator F given by the formula (10) is said to be asso-ciated with �φ . We will write F ∈ Im(X × Y, �φ), if a ∈ Sm+n/2−N/2(X×Y × R

N).Operators from �m,−∞(M, F , E) can be described as Fourier integral opera-

tors associated with the immersed canonical relation G′FN

, which is the image ofGFN

under the mapping GFN→ T ∗M × T ∗M: (γ, ν) �→ (rN(γ, ν), −sN(γ, ν))

[14]. Indeed, consider an elementary operator A: C∞c (U, E|U) → C∞

c (U ′, E|U ′)given by the formula (1) with k ∈ Sm(Ip × Ip × I q × R

q, L(Cr )). It can be rep-resented in the form (10), if we take X = U with coordinates (x, y), Y = U ′ withcoordinates (x ′, y ′), θ = η, N = q, a phase function φ(x, y, x ′, y ′) = (y−y ′)η andan amplitude a = k(x, x ′, y, η). The associated homogeneous canonical relation�φ is the set of all (x, y, ξ, η, x ′, y ′, ξ ′, η′) ∈ T ∗U × T ∗U ′ such that y = y ′, ξ =ξ ′ = 0, η = −η′, that coincides with the intersection of G′

FNwith T ∗U × T ∗U ′.

Moreover, we see that

�m,−∞(M, F , E) ⊂ Im−p/2(M × M, G′FN

;L(E) ⊗ |T (M × M)|1/2).

Since GFNis, in general, an immersed canonical relation, it is necessary to be more

precise in the definition of the classes Im(M × M, G′FN

;L(E) ⊗ |T (M × M)|1/2).This can be done by analogy with the definition of the classes of longitudinalpseudodifferential operators on a foliated manifold given in [4] (see also [14] andthe definition of classes �0,−∞(M, F , E) given above).

Let p be the principal symbol of P , and let �p(t), t ∈ R, be the canonicalrelation in T ∗M × T ∗M defined as

�p(t) = {((x, ξ), (y, η)) ∈ T ∗M × T ∗M : (x, ξ) = f−t (y, η)},


where ft is the Hamiltonian flow of p. It is well-known (cf., for instance, [26]) thateitP is a Fourier integral operator associated with �p(t):

eitP ∈ I 0(M × M, �p(t);L(E) ⊗ |T (M × M)|1/2).

By holonomy invariance of the transverse principal symbol of P , it followsthat �p(t) ◦ GFN

◦ �p(−t) = GFN, and by the composition theorem of Fourier

integral operators (see, for instance, [11]), we have �t(K) = eitP Ke−itP ∈�0,−∞(M, F , E).

Now assume, in addition, that E is the trivial line bundle, the subprincipalsymbol of D2 vanishes, and K ∈ �0,−∞(M, F ) with the principal symbol k ∈S0(GFN

, |T GN |1/2). Denote by LHpthe Lie derivative on C∞(GFN

, |T GN |1/2) bythe vector field Hp. So the function k(t) = F ∗

t (k) ∈ S0(GFN, |T GN |1/2) is the

solution of the equation

dk(t)

dt= LHp

k(t), t ∈ R,

with the initial data k(0) = k. By [10] (cf. also [7, 11]), it follows that, for anyoperator K1 ∈ �0,−∞(M, F ), the operator [P, K1] belongs to �0,−∞(M, F ), and

σ([P, K1]) = 1

iLHp

σ (K1).

Consider any smooth family K(t) ∈ �0,−∞(M, F ), t ∈ R, of operators with theprincipal symbol k(t). Then

dK(t)

dt= i[P, K(t)] + R(t), t ∈ R,

K(0) = K + R0,

where R(t) ∈ �−1,−∞(M, F ), t ∈ R, is a smooth family of operators, and R0 ∈�−1,−∞(M, F ).

Using the fact that �t(K) is the solution of the Cauchy problem

d�t(K)

dt= i[P, �t(K)], t ∈ R,

�0(K) = K,

and the first part of the theorem, we get

K(t) − �t(K) =∫ t

0�t−τ (R(τ)) dτ + �t(R0) ∈ �−1,−∞(M, F ),

and σ(�t(K)) = σ(K(t)) = k(t). �


2.2. THE GENERAL CASE

In this section, we will prove Theorem 1.7 in the general case. Thus, we assume thatD ∈ �1(M, E) is a formally self-adjoint, transversally elliptic operator such thatD2 has the scalar principal symbol and the holonomy invariant transverse principalsymbol.

DEFINITION 2.2. An operator A ∈ �l(M, E) is said to be of order −∞ in someconic neighborhood of N∗F , if, in any regular foliated chart with the coordinates(x, y) ∈ Ip × I q , there exists ε > 0 such that, for any multiindices α and β andfor any natural N , its complete symbol a ∈ Sl(I n × R

n) satisfies the estimate withsome constant CαβN > 0

|∂αξ ∂β

x a(x, y, ξ, η)| < CαβN(1 + |ξ | + |η|)−N,

(x, y) ∈ Ip × I q, (ξ, η) ∈ Rp × R

q, |ξ | < ε|η|.The important fact, concerning to operators of order −∞ in some conic neigh-

borhood of N∗F , is contained in the following lemma [14]:

LEMMA 2.3. If A ∈ �l(M, E) is of order −∞ in some conic neighborhood ofN∗F and K ∈ �m,−∞(M, F , E), then AK and KA are in �−∞(M, E).

Denote by L(D ′(M, E), H∞) (resp. L(H−∞, C∞(M, E))) the space of allbounded operators from D ′(M, E) to H∞ (resp. from H−∞ to C∞(M, E)). Sinceany operator from �−N(M, E) with N > dimM is a trace class operator inL2(M, E), one can easily show the following inclusions

L(D ′(M, E), H∞) ⊂ L1(H−∞, H∞),

L(H−∞, C∞(M, E)) ⊂ L1(H−∞, H∞).(11)

THEOREM 2.4. For any α ∈ R, the operator 〈D〉α = (D2 + I )α/2 can be writtenas

〈D〉α = P(α) + R(α),

where:

(a) P(α) ∈ �α(M, E) is a self-adjoint, elliptic operator with the positive, scalarprincipal symbol and the holonomy invariant transversal principal symbol;

(b) For any K ∈ �∗,−∞(M, F , E), KR(α) ∈ L(H−∞, C∞(M, E)), andR(α)K ∈ L(D ′(M, E), H∞).

Proof. Using the standard construction of parametrix for elliptic operators insome conic neighborhood of N∗F , one gets an analytic family C1(λ), λ �∈ R+, ofoperators from �−2(M, E) such that

C1(λ)(D2 + I − λI) = I − r1(λ), λ /∈ R+, (12)


where r1(λ) ∈ �0(M, E) has order −∞ in some conic neighborhood of N∗F (see[14] for more details). Hence, we have

(D2 + I − λI)−1 = C1(λ) + r1(λ)(D2 + I − λI)−1, λ /∈ R+.

Using the Cauchy integral formula with an appropriate contour � in the complexplane, we get

(D2 + I )α/2 = i

2π

∫�

λα/2−N 〈D〉2N(D2 + I − λI)−1 dλ = P1(α) + R1(α),

with some natural N such that Re α < 2N , where

P1(α) = i

2π

∫�

λα/2−N 〈D〉2NC1(λ) dλ,

R1(α) = i

2π

∫�

λα/2−N 〈D〉2Nr1(λ)(D2 + I − λI)−1 dλ.

In a standard manner (see [14]), one can prove that P1(α) is a transversally ellip-tic operator of class �α(M, E) with the scalar principal symbol and the holonomyinvariant, positive transversal principal symbol.

Let K ∈ �∗,−∞(M, F , E). For any real s, one can write

KR1(α)〈D〉s = i

2π

∫�

λα/2−NK〈D〉2Nr1(λ)〈D〉s(D2 + I − λI)−1 dλ.

By Lemma 2.3, the operator K〈D〉2Nr1(λ) has the smooth kernel and defines abounded operator from H−∞ ⊂ D ′(M, E) to C∞(M, E). Since 〈D〉s(D2 + I −λI)−1 maps H−∞ to H−∞, this implies that the operator KR1(α) is an operator ofclass L(H−∞, C∞(M, E)).

Taking adjoints in (12), we get

(D2 + I − λI)C∗1 (λ) = I − r∗

1 (λ), λ /∈ R+.

It follows that C1(λ) − C∗1 (λ) = C1(λ)r∗

1 (λ) − r1(λ)C∗1 (λ) has order −∞ in some

conic neighborhood of N∗F . Moreover, using the formula

〈D〉2NC1(λ) − C∗1 (λ)〈D〉2N

= 1

λ〈D〉2(〈D〉2(N−1)C1(λ) − C∗

1 (λ)〈D〉2(N−1))〈D〉2 +

+ 1

λ(〈D〉2(N−1)r1(λ) − r∗

1 (λ)〈D〉2(N−1)),

one can prove by induction that 〈D〉2NC1(λ) − C∗1 (λ)〈D〉2N has order −∞ in

some conic neighborhood of N∗F . This implies that the same is true for P1(α) −P ∗

1 (α) = R∗1(α) − R1(α). Combining Lemma 2.3 and duality arguments, we

get that, for any K ∈ �∗,−∞(M, F , E), the operator R1(α)K = (K∗R1(α) +K∗(R∗

1(α) − R1(α)))∗ extends to a bounded operator from D ′(M, E) to H∞.


Let P(α) ∈ �α(M, E) be a self-adjoint, elliptic operator with the positivescalar principal symbol such that the operator P1(α)−P(α) has order −∞ in someneighborhood of N∗F (see also [15]) and R(α) = 〈D〉α/2 −P(α). By Lemma 2.3,for any K ∈ �∗,−∞(M, F , E), the operator K(P (α) − P1(α)) is a smoothingoperator, that immediately completes the proof. �

Let 〈D〉 = P + R be a representation given by Theorem 2.4. Denote by eitP

the strongly continuous group of bounded operators in L2(M, E) generated by theelliptic operator iP . Put also R(t) = eit〈D〉 − eitP .

PROPOSITION 2.5. For any K ∈ �∗,−∞(M, F , E), KR(t), t ∈ R, is a smoothfamily of operators from L(H−∞, C∞(M, E)), and R(t)K, t ∈ R, is a smoothfamily of operators from L(D ′(M, E), H∞).

Proof. By the Duhamel formula, for any K ∈ �∗,−∞(M, F , E) and u ∈H 1(M, E) ⊂ Dom(P ), one can write

KR(t)u = i

∫ t

0eiτP e−iτP KeiτP R ei(t−τ)〈D〉u dτ.

By Theorem 2.1, e−iτP KeiτP ∈ �∗,−∞(M, F , E). Therefore, the operatore−iτP KeiτP R extends to a bounded operator from H−∞ to C∞(M, E). Since eiτP

maps C∞(M, E) to C∞(M, E) and ei(t−τ)〈D〉 is a bounded operator in H−∞, theoperator KR(t) extends to a bounded operator from H−∞ to C∞(M, E).

Using the formula

dn

dtnKR(t) = i

dn−1

dtn−1KPR(t) + inKR〈D〉n−1eit〈D〉, n ∈ N, (13)

one can show by induction that, for any K ∈ �∗,−∞(M, F , E), the function KR(t)

is a smooth function as a function on R with values in L(H−∞, C∞(M, E)). Thesimilar statement, concerning to the operator R(t)K , follows by duality. �

Proof of Theorem 1.7. Let 〈D〉 = P + R be a representation given by Theo-rem 2.4. Let K ∈ �m,−∞(M, F , E). By Theorem 2.1, it follows that the operator�P

t (K) = eitP Ke−itP is in �m,−∞(M, F , E). Moreover, if E is the trivial linebundle, the subprincipal symbol of D2 vanishes, and k ∈ Sm(GFN

, |T GN |1/2) isthe principal symbol of K , then the principal symbol k(t) ∈ Sm(GFN

, |T GN |1/2)

of �Pt (K) is given by k(t) = F ∗

t (k).To complete the proof, it suffices to show that �t(K) − �P

t (K), t ∈ R, is asmooth family of operators of class L1(H−∞, H∞). We have

�t(K) − �Pt (K) = eitP KR(−t) + R(t)Ke−it〈D〉.

Using Proposition 2.5, the fact that the operator eitP takes C∞(M, E) to itselfand (11), we get that eitP KR(−t) belongs to L1(H−∞, H∞). Similarly, usingProposition 2.5, the fact that the operator e−it〈D〉 is bounded in H−∞, and (11),


we get that R(t)K ∈ L(D ′(M, E), H∞) ⊂ L1(H−∞, H∞) and, furthermore,R(t)Ke−it〈D〉 ∈ L1(H−∞, H∞). �

3. Noncommutative Geometry of Foliations

Let (A, H , D) be a spectral triple associated with a compact foliated Riemannianmanifold (M, F ) as in Section 1.4. In this section, we give a description of all theobjects introduced in Section 1.4 for this spectral triple. In particular, we will proveTheorem 1.11.

First, we introduce a notion of scalar principal symbol for an operator of class�m,−∞(M, F , E). Recall that the bundle π∗E on N∗F is holonomy equivariant.Therefore, there is a canonical embedding

i: C∞prop(GFN

, |T GN |1/2) ↪→ C∞prop(GFN

, L(π∗E) ⊗ |T GN |1/2),

which takes k ∈ C∞prop(GFN

, |T GN |1/2) to i(k) = k π∗T . We will identifyC∞

prop(GFN, |T GN |1/2) with its image i(C∞

prop(GFN, |T GN |1/2)) ⊂ C∞

prop(GFN,

L(π∗E) ⊗ |T GN |1/2).We say that P ∈ �m,−∞(M, F , E) has the scalar principal symbol if its prin-

cipal symbol belongs to C∞prop(GFN

, |T GN |1/2). Denote by �m,−∞sc (M, F , E) the

set of all K ∈ �m,−∞(M, F , E) with the scalar principal symbol. For any k ∈C∞

c (G, |T G|1/2), the operator RE(k) is in �0,−∞sc (M, F , E) and its principal sym-

bol σ(RE(k)) is equal to π∗Gk ∈ C∞

prop(GFN, |T GN |1/2) where πG: GFN

→ G isdefined in Section 1.1.

Recall that δ denotes the inner derivation on L(H) defined by 〈D〉 (see Equa-tion (5)). It is easy to see that the class L1(H−∞, H∞) belongs to the domainof δ and is invariant under the action of δ. Moreover, one can easily show thatL1(H−∞, H∞) is an ideal in OP0.

PROPOSITION 3.1. Any operator K ∈ �0,−∞(M, F , E) belongs to OP00. More-

over, for any natural n and for any K ∈ �0,−∞(M, F , E), the operator δn(K)

belongs to �0,−∞(M, F , E) + L1(H−∞, H∞). If K ∈ �0,−∞sc (M, F , E), δn(K)

belongs to �0,−∞sc (M, F , E) + L1(H−∞, H∞).

Proof. Let 〈D〉 = P + R be a representation given by Theorem 2.4. Let δ0

denote the inner derivation on L(H) defined by P :

δ0(T ) = [P, T ], T ∈ Dom δ0 ⊂ L(H).

Let K ∈ �0,−∞(M, F , E). Since the principal symbol of P is scalar and itstransversal principal symbol is holonomy invariant, it is easy to see that δ0(K) isan operator of class �0,−∞(M, F , E), that implies that K belongs to the domainof δn

0 for any natural n.


We will prove by induction on n that any K ∈ �0,−∞(M, F , E) belongs to thedomain of δn for any natural n, and

δn(K) − δn0 (K) ∈ L1(H−∞, H∞).

By Theorem 2.4 and (11), it follows that

δ(K) − δ0(K) = RK − KR ∈ L1(H−∞, H∞).

Now assume that the statement holds for some natural n. Then one can write

δn+1(K) − δn+10 (K) = δ(δn(K) − δn

0 (K)) + Rδn0 (K) − δn

0 (K)R,

that belongs to L1(H−∞, H∞), since δ takes L1(H−∞, H∞) to itself and, byTheorem 2.4, Rδn

0 (K) and δn0 (K)R are in L1(H−∞, H∞).

It remains to note that, by [14], for any K ∈ �0,−∞(M, F , E), the operatorsK〈D〉−1 and 〈D〉−1K are compact operators in L2(M, E). �

Since A = C∞c (G, |T G|1/2) ⊂ �0,−∞(M, F , E), Proposition 3.1 easily im-

plies the following

THEOREM 3.2. For any a ∈ A, the operators a and [D, a] belong to OP0.Moreover, the algebra B generated by δn(a), a ∈ A, n ∈ N is contained in OP0

0.

By Theorem 3.2, it follows that the spectral triple (A, H , D) is smooth. Nextwe will give a description of B and �∗

0 (A).

PROPOSITION 3.3. Any element b ∈ B can written as b = B + T , where B ∈�0,−∞

sc (M, F , E) and T ∈ L1(H−∞, H∞).Proof. By Proposition 3.1, the statement holds for any b of the form δn(a), a ∈

A, n ∈ N. Since L1(H−∞, H∞) is an ideal in OP0, this implies the statement foran arbitrary element of B. �PROPOSITION 3.4. For any natural N , the algebra �∗

0 (A) is contained in�∗,−∞

sc (M, F , E) + OP−N0 .

Proof. Take any P ∈ �∗0 (A) of the form P ∼ ∑+∞

j=0 bq−j 〈D〉q−j with bq−j ∈B. Fix an arbitrary integer j . Let 〈D〉j = P(j) + R(j) be a representation givenby Theorem 2.4. By Proposition 3.3, one can write bj = Bj + Tj , where Bj ∈�0,−∞

sc (M, F , E) and Tj ∈ L1(H−∞, H∞). So we have

bj 〈D〉j = BjP (j) + BjR(j) + Tj 〈D〉j .Here BjP (j) ∈ �

j,−∞sc (M, F , E) (see [14]), BjR(j) ∈ L1(H−∞, H∞) by Theo-

rem 2.4 and Tj 〈D〉j ∈ L1(H−∞, H∞) by the definition of L1(H−∞, H∞). Thus,bj 〈D〉j ∈ �

j,−∞sc (M, F , E) + L1(H−∞, H∞), that completes the proof. �


Now we need the following result on continuity of the principal symbol mapgiven by (2). Let E be a vector bundle on a compact foliated manifold (M, F ). De-note by �0,−∞(M, F , E) the closure of �0,−∞(M, F , E) in the uniform topologyof L(L2(M, E)).

PROPOSITION 3.5. (1) The principal symbol map

σ : �0,−∞(M, F , E) −→ S0(GFN, L(π∗E) ⊗ |T GN |1/2)

extends by continuity to a homomorphism

σ : �0,−∞(M, F , E) −→ S0(GFN, L(π∗E) ⊗ |T GN |1/2).

(2) The ideal Iσ = Ker σ contains the ideal K of compact operators inL2(M, E).

Proposition 3.5 can be proven by an easy adaptation of the proof of analo-gous fact for pseudodifferential operators on compact manifolds (see, for instance,[22, 25]).

Proof of Theorem 1.11. By Proposition 3.4, it follows that the algebra C0 iscontained in �0,−∞(M, F ) + OP−N(H−∞, H∞) for any N and its closure, C0, iscontained in �0,−∞(M, F ) + K . By Proposition 3.5, the principal symbol map σ

induces a map P : S∗A → S0(GFN, |T GN |1/2). By Theorem 1.7, it follows that the

diagram (9) is commutative that completes the proof. �Remark 3.6. Suppose E is a holonomy equivariant vector bundle. Let C∗

E(G)

be the closure of RE(C∞c (G, |T G|1/2)) in the uniform operator topology of

L(L2(M, E)) and C∗r (G) the reduced foliation C∗-algebra (see, for instance, [9]).

By [9], there is a natural surjective projection πE: C∗E(G) → C∗

r (G). The mapπG: GFN

→ G defines a natural embedding C∗r (G) ⊂ S0(GFN

, |T GN |1/2). SinceRE(k) ∈ �0,−∞(M, F , E) for any C∞

c (G, |T G|1/2), C∗E(G) is contained in

�0,−∞(M, F , E). Moreover, the restriction of σ to C∗E(G) coincides with πE . So

the principal symbol map σ provides an extension of πE to �0,−∞(M, F , E). Inparticular, if Iσ = Kerσ coincides with K , then πE is injective, and the holonomygroupoid G is amenable (see, for instance, [1]).

Acknowledgements

The author acknowledges hospitality and support of the Ohio State Universitywhere the work was completed as well as partial support from the Russian Foun-dation for Basic Research, grant no. 04-01-00190. We also thank the referees forcorrections and suggestions.

References

1. Anantharaman-Delaroche, C. and Renault, J.: Amenable Groupoids, Monograph. Enseign.Math. 36, L’Enseignement Mathématique, Geneva, 2000.


2. Block, J. and Getzler, E.: Quantization of foliations, In: Proc. of the XXth Internat. Conf. onDiff. Geom. Methods in Theoretical Physics, June 3–7, 1991, New York City, U.S.A., Vol. 1,World Sci. Publishing, River Edge, NJ, 1992, pp. 471–487.

3. Chen, W. and Ruan, Y.: Orbifold Gromov–Witten theory, In: Orbifolds in Mathematics andPhysics (Madison, WJ, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002,pp. 25–85.

4. Connes, A.: Sur la théorie non-commutative de l’intégration, In: Algèbres d’opérateurs, LectureNotes in Math. 725, Springer, Berlin, 1979, pp. 19–143.

5. Connes, A.: Geometry from the spectral point of view, Lett. Math. Phys. 34 (1995), 203–238.6. Connes, A. and Moscovici, H.: The local index formula in noncommutative geometry, Geom.

Funct. Anal. 5 (1995), 174–243.7. Duistermaat, J. and Hörmander, L.: Fourier intergal operators II, Acta Math. 128 (1972), 183–

269.8. Egorov, Ju. V.: The canonical transformations of pseudodifferential operators, Uspekhi Mat.

Nauk 24(5) (1969), 235–236.9. Fack, T. and Skandalis, G.: Sur les représentations et ideaux de la C∗-algèbre d’un feuilletage,

J. Operator Theory 8 (1982), 95–129.10. Guillemin, V. and Sternberg, S.: Some problems in integral geometry and some related

problems in microlocal analysis, Amer. J. Math. 101 (1979), 915–959.11. Hörmander, L.: The Analysis of Linear Partial Differential Operators IV, Springer, Berlin,

1986.12. Kawasaki, T.: The signature theorem for V -manifolds, Topology 17 (1978), 75–83.13. Kawasaki, T.: The index of elliptic operators over V -manifolds, Nagoya Math. J. 84 (1981),

135–157.14. Kordyukov, Yu. A.: Noncommutative spectral geometry of Riemannian foliations, Manuscripta

Math. 94 (1997), 45–73.15. Kordyukov, Yu. A.: The trace formula for transversally elliptic operators on Riemannian folia-

tions, Algebra i Analiz 12(3) (2000), 81–105; translation from Russian in St. Petersburg Math.J. 12(3) (2001), 407–422.

16. Libermann, P. and Marle, C.-M.: Symplectic Geometry and Analytical Mechanics, Reidel,Dordrecht, 1987.

17. Lichnerowicz, A.: Variétés symplectiques et dynamique attachée à une sous variété, C.R. Acad.Sci. Paris 280 (1975), 523–527.

18. Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées, J. DifferentialGeom. 12 (1977), 253–300.

19. Mardsen, J. E. and Weinstein, A.: Reduction os symplectic manifolds with symmetry, Rep.Math. Phys. 5(1), 121–130.

20. Molino, P.: Riemannian Foliations, Birkhäuser, Boston, 1988.21. O’Neill: The fundamental equations of a submersion, Michigan Math. J. 13 (1996), 459–469.22. Palais, R. S.: Seminar on the Atiyah–Singer Index Theorem, Princeton Univ. Press, Princeton,

1965.23. Reinhart, B. L.: Differential Geometry of Foliations, Springer, Berlin, 1983.24. Satake, I.: The Gauss–Bonnet theorem for V -manifolds, J. Math. Soc. Japan 9 (1957), 464–

492.25. Seeley, R. T.: Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117

(1965), 167–204.26. Taylor, M.: Pseudodifferential Operators, Princeton Univ. Press, Princeton, 1981.27. Trèves, F.: Introduction to Pseudodifferential Operators and Fourier Integral Operators. Vol. 2:

Fourier Integral Operators, Plenum Press, New York and London, 1980.28. Xu, Ping: Noncommutative Poisson algebras, Amer. J. Math. 116 (1994), 101–125.


From Pauli Matrices to Quantum Itô Formula

YAN PAUTRATInstitut Fourier, UMR 5582, BP 74, 38402 Saint-Martin d’Hères Cedex, France

(Received: 19 February 2004; in final form: 5 August 2004)

Abstract. This paper answers important questions raised by the recent description, by Attal, of arobust and explicit method to approximate basic objects of quantum stochastic calculus on bosonicFock space by analogues on the state space of quantum spin chains. The existence of that methodjustifies a detailed investigation of discrete-time quantum stochastic calculus. Here we fully defineand study that theory and obtain in particular a discrete-time quantum Itô formula, which one cansee as summarizing the commutation relations of Pauli matrices.

An apparent flaw in that approximation method is the difference in the quantum Itô formulas,discrete and continuous, which suggests that the discrete quantum stochastic calculus differs fun-damentally from the continuous one and is therefore not a suitable object to approximate subtlephenomena. We show that flaw is only apparent by proving that the continuous-time quantum Itôformula is actually a consequence of its discrete-time counterpart.

Mathematics Subject Classifications (2000): 81S25, 60H05.

Key words: quantum probability, quantum stochastic integrals, Fock spaces, toy Fock space, quan-tum Itô formula.

Introduction

From an early stage in the development of the theory of quantum stochastic calcu-lus on bosonic Fock space, simpler discrete-time versions based on toy Fock spaceshave been considered as a source of inspiration, but only by formal analogy; forexample, such ideas undermine the presentation of the field in [Mey]. Yet it wasnot believed the analogy could be upgraded to a useful tool.

The recent paper [At3] by Attal showed such beliefs to be wrong. That paperdescribes a completely explicit realization of toy Fock space T� as a subspace ofthe usual Fock space � = �sym(L2(R+)) and similarly fundamental noises on T�are expressed in terms of increments of quantum noises on �. These realizationsdepend on some scale; the interesting property here is that, when that scale goesto zero, these objects approximate their continuous-time counterparts. The sim-plicity of the method is surprising, but it should be remarked that its discoveryrelies heavily on the picturesque abstract Itô calculus description of Fock space(see [Mey] or [At2]). Discrete-time objects are naturally simpler than continuous-time ones; here the simplification is a major one since, as should be clear fromthe exposition in this paper, it reduces many problems to finite-dimensional ones.

122 YAN PAUTRAT

There is therefore reasonable hope that continous-time problems can be answeredvia the approximation scheme.

The goal of this paper is to pave the road for systematic application of thisprogram. The first step is a rigorous treatment of discrete-time quantum stochasticcalculus. The state space for that theory is an infinite-dimensional toy Fock space;such a space naturally appears as state space of a chain of two-levels atoms, andin particular, our fundamental noises a+, a−, a◦, a× are just linear combinationsof the usual Pauli matrices. The most natural definitions of integrals

∑hia

εi , ε =

+, ◦, −, ×, turn out to be discrete-time transcriptions of the objects of the Attal–Lindsay theory of quantum stochastic integration (see [A-L]), which extends theearlier versions (developed successively in [H-P], [B-L], [A-M]). This means boththat an important role is played by discrete-time abstract Itô calculus and that ourintegrals enjoy many properties.

Another issue which we address here is the lack, in Attal’s paper, of a relationbetween a quantum stochastic integral on Fock space and the integral representa-tion of its discrete-time approximation. Such a representation exists under fairlygeneral assumptions, as was proved by the author in [Pa1]. What’s more, theserepresentations are explicit and expressed in terms of the discrete-time abstract Itôcalculus. Here we relate the discrete-time Itô calculus to its continuous-time coun-terpart; this allows us to express the integral representation of the approximationof a quantum stochastic integral, in terms of the original integrands.

Such calculations in turn allow us to answer a question which is a probablereason why toy Fock approximation of Fock space quantum stochastic calculus wasnot believed to be a relevant object. Let us describe that question more precisely:on toy Fock space, there is, as we prove it, a quantum Itô formula describing thecomposition of two integrals

∑

i∈N

hεi a

εi

∑

i∈N

kη

i aη

i =∑

i∈N

hεi

(∑

j<i

kη

j aη

j

)

aεi +

+∑

i∈N

(∑

j<i

hεja

εj

)

kη

i aη

i +∑

i∈N

hεi k

η

i aε.η

i ,

where aε.η is actually aεaη, so that it is given by the following table:

� − ◦ + ×− 0 a− a×− a◦ a−◦ 0 a◦ a+ a◦+ a◦ 0 0 a+× a− a◦ a+ a×

which, we recall, is a consequence of the Pauli matrices commutation relations. Wecall that table the discrete time Itô table. On the other hand, it is known that on theFock space, and under some analytical conditions, (see [At1], [A-L], [A-M]) there

FROM PAULI MATRICES TO QUANTUM ITÔ FORMULA 123

is also an Itô formula, of similar form∫ ∞

0Hε

t daεt

∫ ∞

0K

ηt da

ηt =

∫ ∞

0Hε

t

(∫ t

0Kη

s daηs

)

daεt +

+∫ ∞

0

(∫ t

0Hε

s daεs

)

Kηt da

ηt +

∫ ∞

0Hε

t Kηt da

ε.ηt

for the continuous time integrals, but here daε.η is given by

� − ◦ + ×− 0 da− da× 0◦ 0 da◦ da+ 0+ 0 0 0 0× 0 0 0 0

which we call the continuous time Itô table.From the slight difference in the two Itô tables it may seem that there exist

fundamental differences between discrete and continuous stochastic calculus. Bothin order to relieve the approximation scheme from this apparent defect and to showits efficiency we will actually reprove the continuous-time Itô formula from thediscrete-time one and the approximation results – which, we must note, do notdepend on any composition formula.

This paper is organized as follows: in Section 1 we develop a full theory ofquantum stochastic calculus on toy Fock space and recall the statement of the the-orem of representation as discrete quantum stochastic integrals which we use in thesequel. In Section 2 we recall Attal’s approximation method, describe the relationbetween discrete and continuous-time abstract Itô calculus and then compute theintegral representations of approximations of continuous-time quantum stochasticintegrals; we will see that the form of the projections is not as trivial as one wouldexpect. In Section 3 we recover the Itô formula with the associated continuous timeItô table from our approximation scheme and the commutation relations for Paulimatrices.

For notational simplicity, this paper only describes the case of simple toy Fockspace. The case of higher multiplicity Fock spaces is a simple consequence and isdescribed in full detail in the author’s thesis [Pa2], which the interested reader canconsult for a general exposition of the applications of the approximation method.

1. Stochastic Calculus on Toy Fock Space

1.1. DEFINITIONS

Since our main goal is to reproduce as closely as possible the structure of Fockspace continuous calculus, we define many objects by analogy with the continuous-time case. We therefore refer the reader to expositions of the theory of quantumstochastic calculus using abstract Itô calculus, e.g., [A-M] or [At2].

124 YAN PAUTRAT

A basic property of the Fock space � = �sym(L2(R+)) is its Guichardet in-terpretation that makes explicit a unitary equivalence with the space L2(PR+) ofsquare-integrable functions over the set PR+ of finite subsets of R+ (see Section 2,[Gui], or the above references). Similarly, the toy Fock space T� is most naturallydefined as the antisymmetric Fock space over l2(N). Nevertheless, we define it atonce by its ‘Guichardet form’. We denote by P the set of finite subsets of N, thatis, elements of P are either or the form {i1, . . . , in} or the empty set ∅. The toyFock space T� is then defined as the space l2(P ) of square-integrable functions onthe set P of finite subsets of N, that is, T� is the space of all maps f : P �→ C,such that

∑

A∈P

|f (A)|2 < +∞.

When T� is seen as l2(P ), a natural basis arises, that of the indicators 1A

of elements A of P ; we will denote by XA these vectors, and by � the vectorX∅, called the vacuum vector. Every vector f ∈ T� thus admits an orthogonaldecomposition of the form

f =∑

A∈P

f (A)XA.

The toy Fock space has an important property of tensor product decomposition:for any partition

⋃Nj of N, denote by T�Nj

the space l2(PNj) where PNj

is theset of finite subsets of Nj ; then T�Nj

can be identified with a subspace of T� andone has the explicit isomorphism

T� =⊗

j

T�Nj,

where we identify any XA with⊗

j XA∩Nj. We will mainly consider cases where

the Nj ’s are of the form {0, . . . , i − 1} or {i, . . .}; we therefore introduce thenotations

T�i = T�{0,...,i−1}, T�[i = T�{i,...}.

A particular family of elements of T� will be useful in the sequel: it is the familyof exponential vectors. To every u in l2(N) we associate a function on P by

e(u)(A) =∏

i∈A

u(i) for A ∈ P ,

and it can be seen to define a vector in T� by the inequality

n!∑

|A|=n

∣∣∣∣

∏

i∈A

|u(i)|∣∣∣∣

2

�(∑

i�0

|u(i)|2)n

,


but this yields only ‖e(u)‖2 � e‖u‖2and no (simple) formula for 〈e(u), e(v)〉. The

family of exponential vectors is total but contrarily to the case of exponentials ofthe Fock space �sym(L2(R+)), a family of exponentials of distinct functions is notnecessarily linearly independent: consider, for example, the case of u = (0, . . .),v = (1, 0, . . .) and w = (2, 0, . . .), for which e(u) − 2e(v) + e(w) = 0.

Note that the tensor decomposition of an exponential vector is simple: any e(u)

can be decomposed, for example, as e(ui) ⊗ e(u[i ) where ui is the restriction of u

to {0, . . . , i − 1} and u[i is the restriction of u to {i, i + 1 . . .}.

Fundamental operators on T�. One defines for all i ∈ N three operators by theiraction on the basis {XA}:

a+i XA =

{XA∪{i} if i ∈ A,0 otherwise,

a−i XA =

{XA\{i} if i ∈ A,0 otherwise,

(1.1)

a◦i XA =

{XA if i ∈ A,0 otherwise.

These operators are closable, of bounded closures (with norm 1), and we willkeep the same notations for their closures, which we call operators of creation (a+),annihilation (a−) and conservation (a◦). Besides, one should remark that they areof the form Id ⊗ aε

i ⊗ Id in T�i ⊗ T�{i} ⊗ T�[i+1. For notational simplicity wedefine for all i the operator a×

i to be the identity operator.

Relations with Pauli matrices. A more physical description of our frameworkwould start with the following: quantum mechanically speaking, a particle with,for example, two energy levels should be described by the complex vector space ofdimension two: C

2. The customary description for the most important operators ofposition and momentum, which we denote for a few lines by Q and P respectively,is

Q =(

0 11 0

)

, P =(

0 −i

i 0

)

and Q, P satisfy the commutation relation

QP − PQ = 2i Id.

The ∗-algebra generated by Q and P is the whole of the algebra of complex2 × 2 matrices; that algebra is also linearly generated by Id, Q, P and QP . Ifwe denote for consistency Q, P , −iQP by σx , σy , σz respectively, we obtain thefamous Pauli matrices. We therefore have a basis Id, σx , σy , σz with particularalgebraic relations.

126 YAN PAUTRAT

Now if we denote by �, X the canonical basis(

10

),(

01

), there is a more natural

basis for the vector space of 2 × 2 matrices, that is, Id, a+, a−, a◦ with

a+� = X, a−� = 0, a◦� = 0,(1.2)

a+X = 0, a−X = �, a◦X = X,

and we have the relations

a+ = 12(σx − iσy), a− = 1

2(σx + iσy), a◦ = 12(Id − σz), (1.3)

so that the commutation relations for a+, a−, a◦ are straightforward consequencesof those for Pauli matrices. One may wonder why we did not choose the fourthoperator of the canonical basis of linear operators on C

2 instead of the identity; itis only in order to stay as close as can be to the continuous-time case.

Now our toy Fock space is simply the state space for a spin chain, that is, aninfinity of distinguishable particles with two energy levels; the operators aε

i we con-sider are the natural ampliations of the above operators aε, ε = +, −, ◦. More pre-cisely, for any i, aε

i is Id ⊗ aε ⊗ Id in the decomposition T� = T�i ⊗ T�{i} ⊗ T�(i .The relation of the objects we consider with the physically more customary Paulimatrices is therefore clear.

1.2. ABSTRACT ITÔ CALCULUS ON T�

The main difference between Itô calculus on toy Fock space and on regular Fockspace is that predictability should replace adaptability for a simpler transcriptionof the classical results. Therefore we define the (everywhere defined) predictableprojection and gradient at time i ∈ N, of a vector f in T� by

pif (M) = 1M<i f (M),

dif (M) = 1M<i f (M ∪ {i}),where 1M<i is the indicator of the event denoted by M < i which is ‘j < i for allj in M’ (note that ∅ < i for all i).

The above operators are called predictable because for any f ∈ T�, both(pif )i�0 and (dif )i�0 are predictable processes, that is, are sequences of vectorssuch that the ith vector belongs to T�i . In contrast with the continuous time case,there is no definition problem for the di’s as individual operators. We will write, tosimplify notations,

dA = di1 · · · din if A = {i1 < · · · < in},and

d∅ = Id.

The other essential tool for quantum Itô calculus is the abstract Itô integral:


DEFINITION 1.1. A predictable process of vectors (fi)i�0 is said to be Itô-integrable if

∑‖fi‖2 < +∞.

One then defines its Itô integral as the sum of mutually orthogonal terms∑

i

fi Xi.

It can be alternatively described as the vector I (f ) such that

I (f )(M) = f∨M(M − ∨M) and I (f )(∅) = 0,

where ∨M denotes the largest element in the n-uple M .

Let us stress the fact that in fi Xi the product is just a tensor product inT�i ⊗ T�[i thanks to the previsiblity of the process: fi belongs to T�i , Xi belongsto T�[i . The condition for the alternative definition to actually define a square-integrable function of A is easily seen to be the above Itô-integrability condition.

Substituting the equality dif = ∑A<i f (A+i)XA in the chaotic decomposition

of a vector f yields the following results:

PROPOSITION 1.2. Any f ∈ T� admits a unique decomposition of the form

f = f (∅)� +∑

i∈N

dif Xi

and one has the associated isometry formula:

‖f ‖2 = |f (∅)|2 +∑

i∈N

‖dif ‖2.

This decomposition is called the predictable representation of f .

The isometry formula polarizes to the following adjoint relation:⟨∑

i∈N

fiXi, g

⟩

=∑

i∈N

〈fi, dig〉

for all g ∈ T� and all Itô-integrable process (fi)i�0 of vectors of T�.

1.3. QUANTUM STOCHASTIC INTEGRATION ON T�

First of all we have to define predictability of an operator on T�; the followingdefinition is a natural extension of the classical predictability, in the sense that it issatisfied by a ‘quantized’ i-predictable random variable (for the relation betweenquantum and classical stochastic calculus see [At2]).

128 YAN PAUTRAT

DEFINITION 1.3. An operator is i-predictable if it is of the form h ⊗ Id inT�i ⊗ T�[i .

It is clear from this definition that an i-predictable operator is bounded andtherefore can be extended to an everywhere defined operator of the above form.We will therefore always assume predictable operators to be everywhere definedand bounded.

The following lemma can be deduced from the former definition and links ourdefinition with a more algebraic approach which would be a transcription of Attaland Lindsay’s definition in [A-L]:

LEMMA 1.4. A bounded operator h on T� is i-predictable if and only if it satisfiesthe following conditions:

• Its domain Dom h is stable by pi and by all operators dj , j � i.• The following equalities hold on Dom h:

hpi = pih and hdj = djh for all j � i.

Proof. It is clear that an i-predictable operator satisfies the above properties.Conversely, one can prove that the commutation relations in the statement of thelemma are equivalent to the relation

hf (M) = (hpidM∩{i,...})f (M ∩ {0, . . . , i − 1}) (1.4)

for all f in Dom h, all M in P . From this relation one can then show that, for anyvector of the form f ⊗ g in T�i ⊗ T�[i , one has f ∈ Dom h and h(f ⊗ g) =(hf ) ⊗ g. The boundedness of h implies that it is of the form h ⊗ Id. �

We will now define quantum stochastic integrals in discrete time; first remarkthat we wish these integrals to give analogues of predictable representations foroperators. This means that we want integrals to be formally of the form

∑i hi ai ,

where ai denotes an ‘elementary action at time i’. What’s more, we wish to beable to consider the classical case where hi , ai are multiplication operators, andyet the composition hiai should involve no probabilistic interpretation, so that theoperators hi and ai should be tensor-product independent and the composition hiai

be a tensor decomposition in T�i ⊗ T�{i}. We have remarked already that a+i , a−

i ,a◦

i , Id, is a basis for T�{i}; for all these reasons we will consider integrals as seriesof the form

∑i h

εi a

εi where every hε

i is i-predictable.We call predictable process a process (hi)i∈N of operators, such that every hi is

i-predictable.

DEFINITION 1.5. Let (hεi )i∈N be a predictable process. For any ε in {+, −, ◦, ×},

we define the integral of (hεi )i∈N with respect to aε, as the operator series

∑i∈N

hεi a

εi

where this series means that


• its domain Dom∑

i hεi a

εi is the set of all f ∈ T� such that

for all M ∈ P ,∑

i∈N|hε

i aεi f (M)| < +∞,

M �→ ∑i∈N

hεi a

εi f (M) is square-integrable;

• the vector∑

i hεi a

εi f is defined by

(∑

i∈N

hεi a

εi f

)

(M) =∑

i∈N

(hεi a

εi f (M))

for all M in P .

For some of the results to come we will need more restrictive summabilityassumptions; we therefore define restricted integrals:

DEFINITION 1.6. Let (hεi )i∈N be a predictable process. For any ε in {+, −, ◦, ×},

we define the restricted integral∑R

i∈Nhε

i aεi of (hε

i )i∈N with respect to aε, as therestriction of the integral

∑i∈N

hεi a

εi to the set of vectors f in Dom

∑i h

εi a

εi which

are such that

M �→∑

i∈N

|hεi a

εi f (M)|

is a square-integrable function on P .

We have mentioned already the relation between the above described naturaldefinitions of integrals and discrete-time analogues of Attal and Lindsay’s alge-braic definitions of quantum stochastic integrals. One can see from the definitionsof operators aε, ε = +, ◦, −, that

• the quantity a+i f (M) is null if i ∈ M and if i ∈ M then a+

i f (M) = f (M \{i}). Therefore we have for all M in P ,

∑

i

hia+i f (M) =

∑

i∈M

hif (M \ {i}),

and the action of∑

i hia+i gives a ‘discrete Skorohod integral’ of hif ;

• the adapted gradient di equals pia−i , so that for all M in P ,

∑

i

hia−i f (M) =

∑

i ∈M

hif (M ∪ {i});

• the above two remarks and the equality a◦i = a+

i a−i imply that, for all M in

P ,∑

i

hia◦i f (M) =

∑

i∈M

hif (M)

and in each of these equalities it is equivalent for one expression or for the other todefine a summable series (in the case where ε = −) and to define an element of

130 YAN PAUTRAT

l2(P ). It is therefore clear that our integrals (including the case ε = ×) are exactlytranscriptions of Attal and Lindsay’s integrals as defined in [A-L]; in a similar way,the restricted integrals we defined are analogues of their restricted integrals.

In particular these integrals handle just like Attal and Lindsay’s, except that,thanks to the discrete-time framework, the integrands hi are bounded so that one ofthe domain conditions disappears; yet it is, of all conditions, the one least intrinsicto the integral. We take advantage of these analogies to state a few properties ofthese integrals and refer the reader to the proofs in [A-L] instead of reproduc-ing rather tedious computations. Of the properties we state here, the first is adiscrete-time Hudson–Parthasarathy formula for the action of an integral on theexponential domain; the second is an alternative characterization of restricted in-tegrals, in Attal–Meyer form. The third is the famous Itô formula which gives theintegral representation for the composition of two stochastic integrals.

PROPOSITION 1.7 (Hudson–Parthasarathy formulas). Let (hi)i�0 be a predicta-ble process, let ε ∈ {+, ◦, −, ×} and assume an exponential vector e(u) to be inthe domain of the restricted integral

∑iRhε

i aεi . Then for all v in l2(N) one has

⟨

e(u),∑

i∈N

h+i a+

i e(v)

⟩

=∑

i∈N

u(i) 〈e(u1 =i ), h+i e(v1 =i )〉 if ε = +,

⟨

e(u),∑

i∈N

h−i a−

i e(v)

⟩

=∑

i∈N

v(i) 〈e(u1 =i ), h−i e(v1 =i )〉 if ε = −,

⟨

e(u),∑

i∈N

h◦i a

◦i e(v)

⟩

=∑

i∈N

u(i)v(i)〈e(u1 =i ), h◦i e(v1 =i )〉 if ε = ◦,

⟨

e(u),∑

i∈N

h×i a×

i e(v)

⟩

=∑

i∈N

〈e(u), h×i e(v)〉 if ε = ×

and every one of the above series is summable. Here u1 =i (respectively v1 =i)represents the sequence which is equal to u (respectively to v), except for the ithterm, which is null.

The following characterization for restricted integrals can be a most useful tool,especially as it very nicely summarizes the domain conditions for an integral to bedefined.

PROPOSITION 1.8 (Attal–Meyer characterization). Let (hεi )i∈N be four predicta-

ble processes, ε = +, ◦, −, × ; the operator∑ ∑R

i∈Nhε

i aεi is the maximal (in the

sense of domains) operator h satisfying

hf =∑

i∈N

hidif Xi +∑

i∈N

h+i pif Xi +

∑

i∈N

h−i dif +

+∑

i∈N

h◦i dif Xi +

∑

i∈N

h×i pif,


where hi = ∑j<i h

εja

εj and where these equalities mean that

• a vector f is in the domain of h if and only if hidif is Itô-integrable and theother series are Itô-integrable or summable in norm (depending on ε),

• equality holds.

This characterization turns out to be very useful in many proofs; for example,the proof of the following proposition, which seems to reduce to a simple permu-tation of two summations, is made quite painful because of domain considerations.Proposition 1.8 summarizes very nicely these problems so that the proof becomesa (even then tedious) play with commutation relations between integrals and opera-tors pi , di . Notice that in contrast with the continuous-time Attal–Meyer definition(see Section 2), the above is not an implicit definition via a kind of integral equa-tion; thanks to the discrete-time summation, an integral stopped at time i is readilydefined as a finite sum of operators.

The next theorem expresses the composition of two quantum stochastic in-tegrals in integral form. Note that, in the following proposition, the consideredintegrals are restricted ones.

THEOREM 1.9 (Itô formula). Let ε and η be two elements of {+, ◦, −, ×} and(hε

i )i∈N and (kη

i )i∈N be two predictable operator processes on T�. Then the opera-tor

∑

i∈N

Rhiaεi

∑

i∈N

Rkiaη

i −∑

i∈N

Rhεi ki a

εi −

∑

i∈N

Rhikη

i aη

i −∑

i∈N

Rhεi k

η

i aε.η

i (1.5)

is a restriction of the zero process; the symbol aε.η is given by the following table

� − ◦ + ×− 0 a− a×− a◦ a−◦ 0 a◦ a+ a◦+ a◦ 0 0 a+× a− a◦ a+ a×

(1.6)

so that aε.η is simply aεaη.

The comparison between this theorem and its continuous-time analogue will bethe subject of section three.

1.4. INTEGRAL REPRESENTATIONS OF OPERATORS

Here we simply recall results from [Pa1]. In that paper we characterized operatorson T� which can be represented as quantum stochastic integrals, and obtainedexplicit formulas for the integrands. For our purposes here, the most useful resultis the following:

132 YAN PAUTRAT

THEOREM 1.10. Let h be an operator on T� such that all vectors XA belong toDom h ∩ Dom h∗. Then the integral operator with integrands

h+i pi = dihpi,

h−i pi = piha+

i pi,

h◦i pi = diha+

i pi − pihpi

(1.7)

and λ = 〈�, h�〉 is such that

h −(

λ +∑

i�0

h+i a+

i +∑

i�0

h−i a−

i +∑

i�0

h◦i a

◦i

)

is a restriction of the zero process and the set {XA, A ∈ P } is in its domain.

This theorem is not quite enough if we want to consider predictable processesof operators, that is, sequences (hi)i of operators such that hi is i-predictable, andrepresent such a process by

hi =∑

ε=+,◦,−,×

∑

j<i

hεja

εj .

Note that the presence of an integral with respect to a× is unavoidable if we wantthe hε

j ’s to be independent of i. Minor adaptations of the above result allow usto give the following description of representations of processes; moreover, theboundedness of predictable operators simplifies the analytical problems:

COROLLARY 1.11. Let (hj )j∈N be a predictable process of operators on T�.Then for every j the operator hj is equal to

λ +∑

i<j

h+i a+

i +∑

i<j

h−i a◦

i +∑

i<j

h◦i a

◦i +

∑

i<j

h×i a×

i ,

where

h+i pi = dihi+1pi,

h−i pi = pihi+1a

+i pi,

h◦i pi = dihi+1a

+i pi − pihi+1pi,

h×i pi = pi(hi+1 − hi)pi

(1.8)

and λ = 〈�, h0�〉.

2. Approximations of Continuous-Time Integrals

2.1. A REMINDER ON QUANTUM STOCHASTIC CALCULUS

We shall here recall briefly some necessary definitions and results from quantumstochastic calculus on regular Fock space. Again we refer the reader to [Mey] or


[At2] for details on the general framework and on the definition given here ofquantum stochastic integrals.

First of all, thanks to Guichardet’s interpretation we can simply define the Fockspace as � = L2(P ), i.e. the set of functions on the set P of finite subsets ofR+. More precisely, we equip P with a measured space structure using the factthat it is the union of the set of n-tuples and of the empty set. On n-tuples wesimply consider nth-dimensional Borel sets and Lebesgue measure; the empty setis defined to be an atom of mass one. The canonical variable will be denoted by σ ,and the infinitesimal volume element by dσ . Note that we denote the sets of finitesubsets of N or R+ by the same symbol P : the context should always preventconfusion.

The elements of � can be seen as the functions, defined on all increasingsimplices �n = {t1 < · · · < tn} of R+, such that

∑

n

∫

�n

|f (t1, . . . , tn)|2 dt1 · · · dtn < +∞. (2.1)

It is clear from this chaotic representation that � is isomorphic to the chaos spaceof any normal martingale (see [Mey] or [At2] and the references therein), e.g., theBrownian motion, the compensated Poisson process, the Azéma martingales, etc.We shall label as �t the analogous set of functions defined on simplices of [0, t];�t will be canonically included in �.

A particular set of elements in � is relevant, the exponential domain: an expo-nential over a function u ∈ L2(R+) is defined by

E(u)(σ ) =∏

s∈σ

u(s). (2.2)

It is an element of �, as one can see that ‖E(u)‖2 = e‖u‖2. Besides, the exponential

domain E(L2(R+)) is total in � and exponentials are easy to handle so that it is adomain of choice for many proofs.

Abstract Itô calculus. Let us consider for all t the element χt of � defined asfollows:

χt(σ ) ={

1s<t if σ = {s},0 otherwise.

The isomorphism from � to any chaos space sends χt to the Brownian motionat time t when onto the chaos space of Brownian motion, to the Poisson processat time t when onto the chaos space of Poisson process, etc. One can define anintegral of adapted processes (ft )t�0 of elements of � (that is, such that ft ∈ �t

for all t), with respect to the curve (χt )t�0, denoted

I (f ) =∫

ft dχt

134 YAN PAUTRAT

and satisfying

‖I (f )‖2 =∫

R+‖ft‖2 dt, (2.3)

as soon as the latter real-valued integral is finite; the complete construction usesthe isometry property (2.3) for step processes. This integral is called the abstractItô integral.

There is an alternate construction for this integral:

I (f )(σ ) = f∨ σ (σ−), (2.4)

where ∨ σ is the largest element in σ and σ− = σ \ {∨ σ }. The natural conditionsfor this to be well defined can be seen to be the same as above, namely the square-integrability of the process (‖ft‖)t�0.

Let us define the two fundamental operators of abstract Itô calculus on �:

• the adapted projection Pt is defined for all t , as the orthogonal projection onto�t . Explicitly, for any f ∈ �,

Ptf (σ ) = 1σ<tf (σ ); (2.5)

• the adapted gradient is defined by

Dtf (σ ) = 1σ<tf (σ ∪ {t}). (2.6)

As in the discrete-time case, ‘σ < t’ means ‘s < t for all s ∈ σ ’.Substituting (2.6) in (2.4) yields immediately the following analogues to Propo-

sition 1.2:

f = f (∅) +∫

Dtf dχt (2.7)

and

‖f ‖2 = |f (∅)|2 +∫

‖Dtf ‖2 dt. (2.8)

Now notice that we have not been precise in our definition of the operators Dt ;actually it is quite ill-defined in the sense that an individual Dt is not a well-definedoperator. All one can say is, thanks to formula (2.8), that for any f , Dtf is definedfor almost all t .

Quantum stochastic integrals. We shall here define integrals∫ ∞

0Hs daε

s

with respect to the three quantum noises da+, da◦, da− and to time dt , which wedenote by da× for simplicity.


The heuristics of the Attal–Meyer quantum stochastic calculus, which we pres-ent in a simplified way, derives from the fact that the noises, which will turn out tobe differentials of continuous-time fundamental operators, should act just like thefundamental operators of toy Fock space (compare with (1.1)):

• any daεt acts only on �[t,t+dt], which from (2.7) can be seen as ‘generated’ by

� and dχt and• the daε

t are given by the following table:

da+t � = dχt , da−

t � = 0, da◦t � = 0, da×

t � = dt �,

da+t dχt = 0, da−

t dχt = dt �, da◦t dχt = dχt , da×

t dχt = 0.

These heuristics allow us to define integrals∫

Hεs daε

s for adapted processes(Hε

s )s�0, that is, processes of operators such that for almost all s, all f ∈ Dom Hs ,

• the vectors Psf and Duf belong to Dom Hεs for almost all u � s,

• the equalities Hεs Psf = PsH

εs f and Hε

s Duf = DuHεs f hold for almost all

u � s.

In that case, a formal computation (see [A-M] or [Mey]) leads us to give the fol-lowing definition: the integral

∑ε=+,◦,−

∫ ∞0 Hε

s daεs is defined as the only operator

H which satisfies the following equality:

Hf =∫ ∞

0HsDsf dχs +

∫ ∞

0H+

s Psf dχs +

+∫ ∞

0H−

s Dsf ds +∫ ∞

0H ◦

s Dsf dχs (2.9)

with Hs = PsHPs . That is, f is in the common domain of the integrals if andonly if the right-hand side is well defined and equality holds. One can define anintegral

∫ b

aHs daε

s as the integral of the process equal to Hs for s ∈ [a, b] and zerootherwise; one then notices that (2.9) holds equivalently with Hs = ∫ s

0 Hεr daε

r .We also define the integral of an adapted process (H×

s )s�0 as the strong integral∫H×

s ds. The operators aεt are then defined as the integrals

∫ t

0 daεs in the above

sense.We give here as a corollary the formulas of Hudson and Parthasarathy, which

were the cornerstone of the first theory of quantum stochastic integration on Fockspace (described in [H-P]).

Hudson–Parthasarathy formula. Let us consider a quantum stochastic integral H

defined on the exponential domain (see (2.2)). Then the following equality holdsfor all u, v ∈ L2(R+), almost all t ∈ R+:

〈E(u), HE(v)〉 =∫ ∞

0φ(s)〈E(u), Hε

s E(v)〉 ds, (2.10)

136 YAN PAUTRAT

where

φ(s) =

u(s) if ε = +,v(s) if ε = −,u(s)v(s) if ε = ◦,1 if ε = ×.

2.2. EXPLICIT FORMULAS FOR THE PROJECTIONS OF INTEGRALS

In this section we use the embedding of toy Fock space in regular Fock spacedefined by Attal in [At3] and the formulas (1.7) to obtain the projection of anintegral operator in discrete time. Let us first recall the definitions of [At3]: wedenote by P the filter of partitions of R+ partially ordered by inclusion. A genericelement of P is denoted by S = {0 = t0 < t1 < · · ·}, and its mesh size by |S|. Forany S in P we define the following objects on �:

a−i = Id ⊗ a−

ti+1− a−

ti√ti+1−ti

P (1) ⊗ Id,

a+i = Id ⊗ P

(1)i

a+ti+1

− a+ti√

ti+1−ti⊗ Id,

a◦i = Id ⊗ P (1)(a◦

ti+1− a◦

ti)P (1) ⊗ Id,

where P (1) represents the projection on the chaos of order 1, and where the tensordecomposition is meant in �ti ⊗ �[ti ,ti+1] ⊗ �[ti+1 . The space T�(S) ⊂ � isdefined as the closed subspace spanned by the vectors XA = ∏

i∈A Xi for A ∈ P ;it is isomorphic to the toy Fock space, and this isomorphism sends restrictions ofthe above operators aε

i to the operators defined in the previous section. We denoteby ES the projection on the subspace T�(S).

The main relations for our computations are given in the following lemma:

LEMMA 2.1. Let us fix a given partition S = {0 = t0 < t1 < · · · < tn < · · ·}. Onehas for all f ∈ �,

piESf = ESPti f,

diESf = 1√ti+1−ti

ES

∫ ti+1

ti

PtiDtf dt,

and

ES

∫ ∞

0ft dχt =

∑

i�0

1√ti+1−ti

ES

∫ ti+1

ti

Pti ft dtXi.

Proof. The third equality is a consequence of the predictable representationproperty on toy Fock space and of the second equality; the first is straightforward.We therefore prove only the second one.


Since di = pia−i , one has:

diESf = pia−i ESf = 1√

ti+1−tiESPti ((a

−ti+1

− a−ti)f ),

but

(a−ti+1

− a−ti)f =

∫ ∞

ti

(a−t∧ti+1

− a−ti)Dtf dχt +

∫ ti+1

ti

Dtf dt

by the Attal–Meyer formulas, so

diESf = 1√ti+1−ti

piES

(∫ ti+1

ti

Dtf dt

)

= 1√ti+1−ti

ES

(∫ ti+1

ti

PtiDtf dt

)

. �As a corollary, we obtain the following straightforward lemma:

LEMMA 2.2. Let u belong to L2(R+). The projection ESE(u) is again an ex-ponential vector in T� over the function u(i) = (1/

√ti+1−ti)

∫ ti+1ti

u(s) ds. Whenseen as a vector of �, it is not an exponential vector if u = 0, but one has for allti � t < ti+1,

Dte(u) = u(i)√ti+1−ti

e(ui).

Now, if we want to apply Theorem 1.10 to approximations ESHES of inte-grals H , we need to make some assumptions on these integrals.

(HD) The integrals∫ ∞

0 Hεs daε

s and∫ ∞

0 (Hεs )∗ daε′

are well-defined on E(L2(R+))

and all its images by any ES .

Here ε′ is defined by +′ = − −′ = + ◦′ = ◦.

The case of ε = × will be treated later. The assumption (HD) implies in par-ticular, if we denote by H the integral

∫ ∞0 Hs daε

s , that H ∗ equals∫ ∞

0 (Hεs )∗ daε′

on E(L2(R+)) and all its projections by ES . It also implies that the projectionsESHES and ESH ∗

ES are defined on all finite linear combinations of XA’s. Indeed,by Lemma 2.2

ESE(1) = �,

ESE(1[ti ,ti+1]) = √ti+1−ti Xi + �,

ESE(1[ti ,ti+1]∪[tj ,tj+1]) = √ti+1−ti

√tj+1−tj Xi,j +

+ √ti+1−ti Xi + √

tj+1−tj Xj + �

and so on. We therefore apply Theorem 1.10 to obtain the following:

138 YAN PAUTRAT

PROPOSITION 2.3. Let H = ∫Hε

t daεt be a quantum stochastic integral on

� that satisfies the assumptions (HD). Then ESHES has a representation as adiscrete quantum stochastic integral which holds at least on vectors {XA, A ∈ P }and the integrands h+

i , h−i , h◦

i are given by

• for ε = +,

h+i = 1√

ti+1−tiES

∫ ti+1

ti

PtiH+t dt,

h−i = 0,

h◦i = 1

ti+1−tiES

∫ ti+1

ti

PtiH+t (a+

t − a+ti) dt;

• for ε = −,

h+i = 0,

h−i = 1√

ti+1−tiES

∫ ti+1

ti

PtiH−t dt,

h◦i = 1

ti+1−tiES

∫ ti+1

ti

Pti (a−t − a−

ti)H−

t dt;

• for ε = ◦,

h+i = 0,

h−i = 0,

h◦i = 1

ti+1−tiES

∫ ti+1

ti

PtiH◦t dt,

where the equalities are over T�i (considering, for the right-hand side, T�i as asubspace of �ti ) and where all operator integrals are in the strong sense.

In the case where ε = ◦, the integral representation has the exponential domainof T� in its restricted domain.

Remarks:

• Note that for any s < t , (a+t − a+

s )Ps is bounded with norm√

t − s since,on �s , (a+

t − a+s ) is simply multiplication by χt − χs . As a consequence,

Ps(a−t − a−

s ) is also bounded on the exponential domain.• It is rather disappointing that the discrete-time integrals are not defined on a

larger space, for example on the exponential domain of T� in the case whereε = + or −. Let us discuss this phenomenon: notice first that, in any case,

h+i a+

i + h−i a−

i + h◦i a

◦i = ES

∫ ti+1

ti

H εs daε

s ES .


Denote Hi = ∫ ti+1ti

H εs aε

s . For all f ∈ Dom h, all A ∈ P , one has

∑

i

|hεi a

εi f (A)| < +∞ and

∑

i

|h◦i a

◦i f (A)| < +∞, (2.11)

so that the problems will arise from square-summability over A in P . To provethe above claim (2.11), notice first that it is straightforward when ε = + (thesum is finite) and continue with ε = −. The sum

∑i |ESHiESf (A)| is smaller

than

∑

i

1√

(ti1+1−ti1) · · · (tin+1−tin)

∫ ti1+1

ti1

· · ·

· · ·∫ tin+1

tin

|HiESf (s1, . . . , sn)| ds1 · · · dsn (2.12)

if A = {i1, . . . , in}. This is smaller than

(∫ tin+1

tin

‖HsDsESf ‖2 ds

)1/2

+∑

i

∫ ti+1

ti

‖H−s DsESf ‖

which is finite. Besides, ESHiES is h−i a−

i + h◦i a

◦i and the condition∑

i |h◦i a

◦i f (A)| is straightforward since only a finite number of terms in the

sum are nonzero and this implies the condition on∑

i h−i a−

i .

Then one has∑

A∈P |∑i ESHiESf (A)|2 < +∞ since it is dominated by‖ ∑

i HiESf ‖2. Therefore one deduces that∑

A∈P

|∑

i

(hεi a

εi + h◦

i a◦i )f (A)|2 < +∞, (2.13)

but there is no reason why one should have

∑

A∈P

∣∣∣∣

∑

i

h+i a+

i f (A)

∣∣∣∣

2

< +∞ and∑

A∈P

∣∣∣∣

∑

i

h◦i a

◦i f (A)

∣∣∣∣

2

< +∞;

choose, for example, ε = + and f = E(u) to see in detail what happens. Forti � t < ti+1 one has

Pte(u) = e(ui) + u(i)e(ui)χt − χti

ti+1−ti(2.14)

which implies (2.13) by the Attal–Meyer formulation but this does not implythat

∫ ∞

0‖H+

t e(ui)‖2 dt +∫ ∞

0

∥∥∥∥H+

t u(i)e(ui)χt − χti

ti+1−ti

∥∥∥∥

2

dt < +∞.

140 YAN PAUTRAT

One can on the other hand obtain the definiteness of the integral∑

i h◦i a

◦i on

the exponential domain from the fact that for ti � t < ti+1,

u(i)√ti+1−ti

H ◦t e(ui) = H ◦

t Dte(u).

The equality (2.14) is also the reason for the surprising presence of a a◦ integralin the projection of an integral with respect to aε when ε is + or −. We prove nowthat that parasite integral vanishes in the limit; yet, since we do not know if thatintegral is well-defined beyond the linear span of {XA, A ∈ P }, we have to provethat result on a subdomain of E(L2(R+)) such that its projections ES belong tothat linear span. The set of exponential vectors of square-integrable functions withcompact support is such a subdomain.

LEMMA 2.4. Let H = ∫ ∞0 Hs daε

s be an integral that satisfies the assumptions(HD) with ε = + or −. Then the parasite a◦ integral which arises in the projectionESHES vanishes, in the sense that the net

(

ES

∑

i∈N

h◦i a

◦i ES

)

S∈P

tends to zero in the w*-topology on the set of exponentials of functions with com-pact support.

Proof. Take, for example, ε = +; then one can see from the discrete Hudson–Parthasarathy equation (Proposition 1.7) that

⟨

E(u), ES

∑

i

h◦i a

◦i ESE(v)

⟩

=∑

i

u(i)v(i)〈e(u1 =i), h◦i e(v1 =i)〉

=∑

i

u(i)√ti+1−ti

∫ ti+1

ti

〈e(ui), H+t Pt (e(vi+1) − e(vi))〉 dt

=∑

i

∫ ti+1

ti

〈DtESE(u), H+t Pt (e(vi+1) − e(vi))〉.

Now, by the assumptions (HD) we know that∫ ∞

0‖(H−

t )∗DtESE(u)‖dt < +∞

whereas ‖e(vi+1) − e(vi)‖ is of order v(i), which is smaller than√∫ ti+1

ti

‖v(t)‖2 dt


and converges to zero uniformly in i. �Proof of Proposition 2.3. Let us prove, for example, the case ε = +. Let us

consider the action of h+i , h−

i , h◦i on a vector XA with A < i. One has by (1.7) and

Lemma 2.1,

h+i XA = diESHXA

= 1√ti+1−ti

ES

∫ ti+1

ti

PtiDtHXA dt

and the Attal–Meyer equations yield DtHXA = H+t XA, and we obtain the expres-

sion of h+i . The value of h−

i is easily computed:

h−i XA = piESHa+

i XA

= ESPtiXA∪{i}= 0,

because, again by straightforward application of the Attal–Meyer formula,

HXA∪{i} =∫ ti+1

ti

Hs

XA√ti+1−ti

dχs +∫ ti+1

ti

H+s XA

χs − χti√ti+1−ti

dχs.

Finally,

h◦i XA = diESHa+

i XA − piESHXA

= 1√ti+1−ti

ES

∫ ti+1

ti

PtiDtHXA∪{i} dt − ESPtiHXA.

From the above computation,

PtiDtHXA∪{i} = 1√ti+1−ti

PtiHtXA + 1√ti+1−ti

PtiH+t (XA(χt − χti ));

now we have

PtiHtXA − PtiHXA = 0

for t � ti and

H+t (XA(χt − χti )) = H+

t (a+t − a+

ti)XA.

The proof is complete. The other two cases are treated exactly in the same way.The definiteness of the restricted integral on the exponential domain when ε = ◦was proved in the discussion following the statement of Proposition 2.3. �

Suppose now that we want to project an integral H = ∫H×

s da×s ; it is poss-

ible to compute an integral representation for ESHES , but the coefficients do not

142 YAN PAUTRAT

express simply in terms of (H×t )t . The same is true if we compute the integral

representation of the process (ESHti ES)i . The reason is the following: the repre-sentation of H as

∫ ∞0 H×

s ds is not unique; if we compute projections accordingto our schemes, be it projections of the process or projections of the operator only,we compute an unique representation, that is, we try to compute a representationwith more information than the original one, so that one can not relate explicitlythe coefficients of the projection to the process (H×

s )s∈R+ . This is why it willbe more convenient to take a completely different approach to project integrals∫ ∞

0 H×s da×

s . We only state the following proposition, which shows an alternativeway to project integrals with respect to time and is proven immediately by anAttal–Meyer argument:

PROPOSITION 2.5. Let H = ∫ +∞0 H×

s da×s be an integral in � which satisfies

the assumptions (HD). Then ESHES is equal on the exponential domain to therestricted integral

∑i�1R h

′×i a×

i , where

h′×i = ES

∫ ti

ti−1

H×t dt ES .

We should emphasize here the fact that the representation given above is not acontradiction to the formulas (1.8): it is just another consequence of the fact that indiscrete-time also, the representation of one operator h as h = ∑

ε=+,◦,−,×∑

i hεi a

εi

is not unique.

3. Convergence of the Itô Table

3.1. A PROOF OF THE ITÔ FORMULA BY APPROXIMATION

In this subsection we want to prove that the Itô formula for continuous-time quan-tum stochastic integrals is a limit of the one for discrete-time integrals; to achievethis we actually reprove the quantum Itô formula for regular semimartingales as de-fined in [At1], using nothing but our approximation scheme and the Itô formula ontoy Fock space. Throughout this section we will make the following assumptionson the operator integrals

H =∫ ∞

0Hε

t daεt :

(HS)

(1) the integrands Hεt are bounded operators such that t �→ ‖Hε

t ‖ is:

• square-integrable if ε = + or −,

• integrable if ε = ×,

• essentially bounded if ε = ◦;(2) H is a bounded operator on �.


Notice that these assumptions are those made for regular semimartingale processesas defined in [At1].

Let us fix the notations to be used in the rest of the paper: we will consider twointegrals

H =∫ ∞

0Hε

s daεs and K =

∫ ∞

0Kη

s daηs

which satisfy the assumptions (HS); ε and η can take the values +, −, ◦ or ×.The projections ESHES , ESKES will be denoted by h, k, respectively. In the casewhere ε or η is different from ×, the processes (hε

i )i∈N, (kη

i )i∈N and (h◦i )i∈N, (k◦

i )i∈N

are as defined in Proposition 2.3; we will discuss again later the case of ε = ×. Ifε is + or − then we have seen that h is equal to

h =∑

i

hεi a

εi +

∑

i

h◦i a

◦i ;

we then denote by h◦ the integral∑

i h◦i a

◦i which we will usually call the ‘parasite’

term. As before, hj will denote the integral stopped at time j

hj =∑

i<j

hεi a

εi +

∑

i<j

h◦i a

◦i ;

and h◦j the integral

h◦j =

∑

i<j

h◦i a

◦i .

The proof will be done in four steps. In the first step, we will discuss the validityof obtained integrals. We show that they are valid in the restricted sense on thewhole of Fock space, when ε = +, ◦ or −. This will allow us to apply the discreteItô formula freely. To simplify further proofs, we give an alternative descriptionof the projection where ε = × integrals. In the second step, we show that theunwanted a◦ integrals that appear when projecting integrals with respect to a+ ora− vanish, as well as the terms they create when two projections are composed. Inthe third step we prove that, asymptotically, one can compute the composition oftwo projections using the continuous Itô table. Last, we show that the remainingdiscrete-time integrals obtained after composition do converge to the continuous-time integrals we are looking for.

Validity of the discrete integrals. First let us assume that ε = ×. With suchassumptions, it is straightforward from general stochastic integration theory on �

that H = ∫ ∞0 Hε

t daεt is the strong limit of the

∫ T

0 Hεt daε

t as T goes to infinity, withuniform norm estimates. As a consequence, H is the strong sum of all

∫ ti+1ti

H εt daε

t

and ESHES is the strong sum of all

ES

∫ ti+1

ti

H εt dt ES = h+

i a+i + h−

i a−i + h◦

i a◦i .

144 YAN PAUTRAT

The following lemma, which will allow us to use full power of the Attal–Meyer for-mulation, proves that the obtained integral representations are valid in the restrictedsense on the whole of T�.

LEMMA 3.1. Let ε equal +, − or ◦ and the integral∫ ∞

0 Hεs daε

s satisfy the as-sumptions (HS). Then the integral

∑i h

εi a

εi associated to ESHES has the whole of

T� for restricted domain.Proof. By the Attal–Meyer characterization of restricted integrals it is enough

to prove that for all f in �, the vector ESf of T� is such that∑

i ‖hidiESf ‖2 and∑

i ‖h+i piESf ‖2

,∑

i ‖h−i diESf ‖ or

∑i ‖h◦

i diESf ‖2, depending on ε, are finite.This is obtained from the fact that ‖hi‖ and ‖h◦

i ‖ are uniformly bounded and that

‖h±i ‖2 �

∫ ti+1

ti

‖Hs‖2 ds

is square-integrable. �Now let us consider the case when ε = ×. The representation of ESHES

given by Proposition 2.5 would eventually lead us to compare∫ titi−1

integrals to∫ ti+1ti

integrals. To avoid this problem we give the following alternative description:

PROPOSITION 3.2. Let H = ∫ ∞0 H×

s ds be an operator satisfying (HS). ThenESHES is the strong limit on T� of the series

∑i h

×i a×

i , where

h×i pi = ES

∫ ti+1

ti

PtiH×t dt ESpi.

Proof. The series∑

i h×i a×

i has clearly T� as restricted domain by the Attal–Meyer characterization. We are going to prove that, for all u in L2(R+),

(∑

i

h×i a×

i −∑

i

h′×i

)

ESE(u)

converges to zero in norm. Since the considered operators are norm-bounded, thiswill prove strong convergence of T�. The above quantity is

∑

i�0

h×i e(ui)e(u[i ) −

∑

i�0

h′×i+1e(ui+1)e(u[i+1)

=∑

i

(

ES

∫ ti+1

ti

PtiH×t e(ui) dt ⊗ e(u[i )−

− ES

∫ ti+1

ti

H×t e(ui+1) dt ⊗ e(u[i+1)

)

,


and replacing e(ui), e(ui+1) in the two integrals by Pte(u) creates an error termwhich smaller than

∑

i

|u(i)|∫ ti+1

ti

‖H×s ‖ ds‖e(u)‖ +

∑

i

|u(i + 1)|∫ ti+1

ti

‖H×s ‖ ds‖e(u)‖,

so that it converges to zero. A similar estimate holds for the substitution of e(u[i ),e(u[i+1) by (� + χti+1 − χt/

√ti+1−ti u(i)) ⊗ e(u[i+1). What we end up with, after

these substitutions, is

∑

i

ES

∫ ti+1

ti

(Pti − Id)H×t Pte(u) dt.

This can be rewritten as

ES

∫ ∞

0(Pti − Id)H×

t Pte(u) dt,

where the i is actually a i(t). Now Lebesgue’s theorem applies and shows that theabove integral tends to zero with the mesh size of the partition. �

Remark. Notice that a projection ES

∫H×

s da×s ES will, in our scheme, be com-

posed only with bounded operators. Besides, we will from now on only be in-terested in weak convergences so that one can always consider adjoint relations.Because of that, we will systematically replace the projection ESHES of an integralH = ∫

H×s da×

s by the series described in Proposition 3.2.

The vanishing of parasite terms. In this paragraph we show that the unwanted a◦from the projection, as well as the terms they induce after composition, vanish inthe limit.

PROPOSITION 3.3. Let ε, η ∈ {+, −, ◦, ×} and let H, K be two operator inte-grals satisfying the assumptions (HS). Then the net

(

ESHES ESKES − ES〈e(u)〉,∑

i

hεi a

εi

∑

i

kη

i aη

i ES

)

S∈P

tends to zero on the exponential domain in the w*-topology.Proof. If both ε and η are ◦, there is nothing to do but recall that ESHES and

ESKES converge strongly on � and are uniformly bounded in norm. If one of ε,η is ×, the corresponding projection can be immediately replaced by the integralwith respect to a× thanks to the emphasized consequence of Proposition 3.2.

To work out the other cases notice that

hk = ((h − h◦) + h◦)((k − k◦) + k◦) if ε and η are both + or −,

hk = ((h − h◦) + h◦)k if, for example, ε only is + or − .

146 YAN PAUTRAT

Besides, h − h◦ = ∑i h

εi a

εi , so that one only has to show that h◦k, hk◦ and h◦k◦

tend to zero in our weak sense. Thanks to adjointness properties the proof reducesto proving that

h◦k converges to zero for ε = −, + and any η (3.1)

and

h◦k◦ converges to zero when ε and η are both + or −, (3.2)

where convergence is meant weakly on the exponential domain.We first prove (3.1); for this let us state the simple estimate

‖h◦i e(ui)‖ � 1√

ti+1−ti

∫ ti+1

ti

‖Hεt ‖ dt‖e(ui)‖, (3.3)

obtained by using the first remark after Proposition 2.3. This estimate implies

‖h◦i e(ui)‖ � ‖u‖

√∫ ∞

0‖Hε

t ‖2 dt exp‖u‖2/2. (3.4)

Observe that

〈e(u), h◦ke(v)〉 = 〈h◦∗e(u), ke(v)〉

and that, since k = ESKES is bounded and (h◦)∗e(u) is uniformly bounded by(3.4), one can replace e(v) by anything which tends to it in norm with the mesh size|S|. One can approximate KE(v) by a linear combination of exponential vectors;let us suppose for simplicity that KE(v) is approximated by a single vector E(w).Then, since

‖ke(v) − e(w)‖ = ‖ESE(w) − ESKESE(v)‖� ‖ESE(w) − ESKE(v)‖ + ‖ESKE(v) − ESKESE(v)‖� ‖KE(v) − E(w)‖ + ‖K‖‖E(v) − ESE(v)‖,

one can replace ke(v) by e(w). Now our assumption reduces to showing that〈e(u), h◦e(w)〉 tends to zero with |S|. It is equal to

∑

i

u(i)v(i)〈e(ui), h◦i e(wi)〉〈e(u[i+1), e(v[i+1)〉

=∑

i

u(i)v(i)

ti+1−ti

∫ ti+1

ti

〈e(ui), Pti (a−t − a−

ti)H−

t e(wi)〉 dt〈e(u[i+1), e(v[i+1)〉

if, for example, ε = − (the case ε = + is proved by dual computations). By anestimate similar to (3.4) we have

|〈e(u), h◦e(w)〉| � ‖u‖‖w‖exp‖u‖exp‖w‖√

supi

∫ ti+1

ti

‖H−t ‖2

dt


and the last term converges to zero as the mesh size of the partition goes to zero.To prove (3.2) let us write

h◦k◦ =∑

i

h◦i k

◦i a

◦i +

∑

i

h◦i k

◦i a

◦i +

∑

i

h◦i k

◦i a

◦i

using the discrete time Itô formula. That implies

〈e(u), h◦k◦e(v)〉 =∑

i

u(i)v(i)〈e(ui), (h◦i k

◦i + h◦

i k◦i + h◦

i k◦i )e(vi)〉

up to uniformly bounded factors 〈e(u[i+1), e(v[i+1)〉. For the sake of clarity, we willignore them and all their avatars from now on. Using such estimates as (3.3) andthe fact that h◦

i , k◦i are bounded with norms � ‖H‖, ‖K‖, one has a majoration

of |〈e(u), h◦k◦e(v)〉| by three terms of the kind

∑

i

|u(i)v(i)|√∫ ti+1

ti

‖Kt‖2 dt

and since the series∑

u(i)v(i) is convergent, this tends to zero with the mesh sizeof the partition. �We now move on to the third step of our proof, contained in the following propo-sition.

PROPOSITION 3.4. With the assumptions of Proposition 3.3, the net(

ESHES ESKES − ES

(∑

i

hεi kia

εi +

∑

i

hikη

i aη

i +∑

i

hεi k

η

i aε.η

i

)

ES

)

S∈P

,

where ε.η is computed using formally the continuous Itô table, tends to zero on theexponential domain in the w*-topology.

Proof. All that is left to prove is that

for (ε, η) = (+, −), one has∑

i

h+i k−

i a◦i →

|S|→00 (3.5)

and

for (ε, η) = (−, +), one has∑

i

h−i k+

i a◦i →

|S|→00 (3.6)

plus the convergence to zero in all cases involving an integral with respect to ×.The proofs of (3.5), (3.6) are the same; let us prove, for example, (3.5):

∣∣∣∣〈e(u),

∑

i

h−i k+

i a◦i e(v)〉

∣∣∣∣ �

∑

i

|u(i)||v(i)|‖h−∗i e(ui)‖‖k+

i e(vi)‖

148 YAN PAUTRAT

up to a constant factor, and since

‖h−∗i e(ui)‖ �

√∫ ti+1

ti

‖H−∗t ‖2

dt‖e(ui)‖

with a similar estimate for ‖k+i e(vi)‖, one concludes as before.

Now in the case where ε, for example, is ×, one writes the usual equalities

〈e(u), aη(h×kη)e(v)〉=

∑

i∈N

〈e(ui), h×i k

η

i e(vi)〉〈e(u[i ), e(v[i )〉

=∑

i∈N

∫ ti+1

ti

∫ ti+1

ti

〈e(ui), H×t Kη

s e(vi)〉〈e(u[i ), e(v[i )〉 dt ds

(keep in mind that ×.η = η in all cases), and this is dominated by the followingquantities:

• ∑i

∫ ti+1ti

‖H×t ‖dt

∫ ti+1ti

‖K×t ‖dt if η = ×,

• ∑i

√ti+1−ti

∫ ti+1ti

‖H×t ‖dt

√∫ ti+1ti

‖Kηt ‖2 dt if η = +, −,

• ∑i (ti+1−ti)

∫ ti+1ti

‖H×t ‖dt ‖K◦‖∞ if η = ◦,

where all majorations are up to constant factors. In all three cases the summed termin the majorant is a summable term multiplied by a vanishing one. �

We now apply these results to prove the final result:

THEOREM 3.5 (Itô formula in continous time). Let H = ∫ ∞0 Hε

s daεs and K =∫ ∞

0 Kηs da

ηs be two continuous-time integrals satisfying the assumptions (HS). Then

the following equality holds on �:∫ ∞

0Hε

t daεt

∫ ∞

0K

ηt da

ηt =

∫ ∞

0Hε

t Kt daεt +

∫ ∞

0HtK

ηt da

ηt +

∫ ∞

0Hε

t Kηt da

ε.ηt ,

where aε.η is computed using the continuous Itô table.

Remark. Let us repeat that this reproves the Itô formula on regular Fock spaceknowing nothing but its counterpart on the toy Fock space.

Proof of Theorem 3.5. We will prove that for any u, v ∈ L2(R+) one has⟨

E(u),

∫ ∞

0Hε

t daεt

∫ ∞

0K

ηt da

ηt E(v)

⟩

=⟨

E(u),

(∫ ∞

0Hε

t Kt daεt +

∫ ∞

0HtK

ηt da

ηt +

∫ ∞

0Hε

t Kηt da

ε.ηt

)

E(v)

⟩

.


By Proposition 3.4, it suffices to show that⟨

e(u),∑

i

hεi kia

εi e(v)

⟩

−→⟨

E(u),

∫ ∞

0Hε

s Ks daεs E(v)

⟩

,

⟨

e(u),∑

i

hikη

i aη

i e(v)

⟩

−→⟨

E(u),

∫ ∞

0HsK

ηs daη

s E(v)

⟩

,

⟨

e(u),∑

i

hεi k

η

i aε.η

i e(v)

⟩

−→⟨

E(u),

∫ ∞

0HεKη daε.η

s E(v)

⟩

but one can see that the previous propositions apply to the integrals∫

Hεs Ks daε

s ,∫HsK

ηs da

ηs ,

∫Hε

s Kηs da

ε.ηs so that one also has that

⟨

e(u),∑

i

(H εK)εi a

εi e(v)

⟩

−→⟨

E(u),

∫ ∞

0Hε

s Ks daεs E(v)

⟩

,

⟨

e(u),∑

i

(HKη)η

i aη

i e(v)

⟩

−→⟨

E(u),

∫ ∞

0HsK

ηs daη

s E(v)

⟩

,

⟨

e(u),∑

i

(H εKη)ε.η

i e(v)

⟩

−→⟨

E(u),

∫ ∞

0Hε

s Kηs daε.η

s E(v)

⟩

,

where (HKη)η

i , (HεK)εi are the integrands associated by Proposition 2.3 (or by

Proposition 3.2) to the integral∫

HsKηs da

ηs , etc. It suffices then to prove that

⟨

e(u),∑

i

(hεi ki − (HεK)ε

i )aεi e(v)

⟩

−→ 0, (3.7)

⟨

e(u),∑

i

(hikη

i − (HKη)η

i )aη

i e(v)

⟩

−→ 0, (3.8)

⟨

e(u),∑

i

(hεi k

η

i − (HεKη)ε.η

i )aε.η

i e(v)

⟩

−→ 0. (3.9)

The convergences (3.7) and (3.8) derive one from another by adjointness. Letus prove the different cases one by one.

Proof of (3.7). First consider (3.7) in the case ε = − or +: let us take, forexample, ε = +.

〈e(u), aε(hεk − (HεK)εi )e(v)〉 =

∑

i

u(i)〈e(ui), (h+i ki − (H+K)+

i )e(vi)〉.

The above quantities are equal to

∑

i

u(i)

⟨

e(ui),1√

ti+1−ti

∫ ti+1

ti

Pti (H+t ki − H+

t Kt )e(vi) dt

⟩

150 YAN PAUTRAT

=∑

i

u(i)√ti+1−ti

∫ ti+1

ti

〈H+∗t e(ui), (ki − Kt)e(vi)〉 dt.

So the norm of the left-hand side is smaller than

∑

i

|u(i)|√ti+1−ti

∫ ti+1

ti

‖H+∗t ‖‖(ki − Kt)e(vi)‖dt

�∑

i

|u(i)|√∫ ti+1

ti

‖Ht‖2‖(ki − Kt)e(vi)‖2 dt

� ‖u‖2l2

√∫ ∞

0‖H+

t ‖2‖(ki − Kt)e(vi)‖2 dt (3.10)

by repeated use of the Cauchy–Schwarz formula and convenient erasing of constantterms. The index i in the last line is actually a i(t).

But, since ki = ESKti ES ,

‖(ki − Kt)e(vi)‖ � ‖Kti e(vi)‖ + ‖Kte(vi)‖.If η = +, ◦, −, then (Kt) is an operator martingale, so, since ti � t with e(vi) ∈�ti ,

‖(ki − Kt)e(vi)‖ � ‖ESKti e(vi)‖ + ‖Kte(vi)‖� 2‖Ke(vi)‖� 2‖K‖‖e(v)‖.

A majoration of the same kind is immediately obtained in the case η = × since‖Kt‖ �

∫ ‖K×s ‖ ds. One can then apply Lebesgue’s dominated convergence theo-

rem to the integral in (3.10). Besides,

‖(ki − Kt)e(vi)‖ � ‖(ESKti − Kti )e(v)‖ + ‖(Kti − Kt)Pti ESE(vti )‖and both terms on the right-hand side tend to zero a.e.

We now consider (3.7) in the case ε = ◦: consider the quantity∑

i

u(i)v(i)〈e(ui), (h◦i ki − (H ◦K)◦

i )e(vi)〉,

where we forget once again the last factor. It is equal to

∑

i

u(i)v(i)

ti+1−ti

∫ ti+1

ti

〈e(ui), H◦t (ki − Kt)e(vi)〉 dt

=∫ ∞

0

u(i)v(i)

ti+1−ti〈e(ui), H

◦t (ki − Kt)e(vi)〉 dt.


The bracket is uniformly bounded, and t �→ u(i)/√

ti+1−ti , v(i)/√

ti+1−ti whereonce again i is actually a i(t), tend to u, v in L2(R+) by the martingale convergencetheorem. One can therefore consider, instead of the above, the quantity

∫ ∞

0u(t)v(t)〈e(ui), H

◦t (ki − Kt)e(vi)〉 dt

so that one can now apply Lebesgue’s theorem in the same way as in the previouscase.

Now turn to (3.7) in the case ε = ×: we have∑

i

〈e(ui), (h×i ki − (H×K)×

i )e(vi)〉

=∑

i

∫ ti

ti−1

〈H×∗t e(ui), (ESKti − Kt)e(vi)〉 dt,

and we conclude as before. �Proof of (3.9). Let us now prove (3.9). The following lemma will be most useful:

LEMMA 3.6. One has the following estimates:

• if η = +, then kη

i aη

i e(vi+1) = kη

i e(v1 =i)Xi , and

‖kη

i aη

i e(vi+1)‖ �√∫ ti+1

ti

‖Kηt ‖2 dt ‖E(v)‖;

• if η = −, then kη

i aη

i e(vi+1) = v(i)kη

i e(v1 =i) and

‖kη

i aη

i e(vi+1)‖ �√∫ ti+1

ti

‖Kηt ‖2 dt |v(i)|‖E(v)‖;

• if η = ◦, then kη

i aη

i e(vi+1) = v(i)kη

i e(v1 =i)Xi and

‖kη

i aη

i e(vi+1)‖ � sup‖K◦s ‖ |v(i)|‖E(v)‖;

• if η = ×, then k×i a×

i e(vi+1) = (k×i e(vi))(� + v(i)Xi) and

‖kη

i aη

i e(vi+1)‖ �∫ ti+1

ti

‖K×t ‖ dt (1 + |u(i)|).

First let us treat the case where one of ε or η is ×; we take, for example, ε = ×.In this case we have to show that

⟨

e(u),∑

i

h×i k

η

i aη

i e(v)

⟩

152 YAN PAUTRAT

vanishes when the mesh size of the partition tends to zero. But that quantity issmaller in norm than

∑

i

‖(h×i )∗e(u)‖‖kη

i aη

i e(v)‖

which in turn is smaller than a constant times

∑

i

∫ ti+1

ti

‖H×∗t ‖ dt‖kη

i aη

i e(vi+1)‖. (3.11)

From Lemma 3.6, we obtain that whatever η, (3.11) is a sum of terms of the form∫ ti+1ti

‖H×∗t ‖ times a term that vanishes uniformly in i with the mesh size of the

partition.There are four nontrivial cases left: (ε, η) equal to (−, +), (◦, ◦), (◦, +) and

(−, ◦). The last two cases have similar proofs; let us prove them first. We thereforeconsider (3.9) in the case (ε, η) = (−, ◦) or (◦, +): take, for example, (ε, η) =(◦, +). What we want to prove is that

⟨

e(u),∑

i

(h◦i k

−i − (H ◦H+)+

i )a+i e(v)

⟩

−→|S|→0

0.

This quantity is equal to

∑

i

u(i)

⟨

e(ui), (h◦i k

+i − 1√

ti+1−ti

∫ ti+1

ti

PtiH◦t K+

t dt)e(vi)

⟩

=∑

i

u(i)

⟨

e(ui),1√

ti+1−ti

∫ ti+1

ti

H ◦t

(1√

ti+1−tik+

i − K+t

)

dte(vi)

⟩

,

hence the norm of the left-hand side is, up to a factor term depending only on u, v,

smaller than:

∑

i

|u(i)|√ti+1−ti

∫ ti+1

ti

‖H ◦t ‖

∥∥∥∥

(1√

ti+1−tik+

i − K+t

)

e(vi)

∥∥∥∥ dt

� sup‖H ◦t ‖

∑

i

|u(i)|√

∫ ti+1

ti

∥∥∥∥

(1√

ti+1−tik+

i − K+t

)

e(vi)

∥∥∥∥

2

dt .

Since e(vi+1) − e(vi) = v(i)e(vi)Xi , substituting e(vi) with e(vi+1) creates anerror term which is smaller than

(∑

i

∫ ti+1

ti

∥∥∥∥

(1

ti+1−ti

∫ ti+1

ti

K+s ds − K+

t

)

v(i)e(vi)Xi)

∥∥∥∥

2

dt

)1/2

�(

2∑

i

∫ ti+1

ti

(1

(ti+1−ti)2

(∫ ti+1

ti

‖Ks‖ ds

)2

+ ‖Kt‖2

)

|v(i)| dt

)1/2


up to a constant factor; but that is smaller by the Cauchy–Schwarz inequality than

(∑

i

∫ ti+1

ti

(1

ti+1−ti

∫ ti+1

ti

‖Ks‖2 ds + ‖Kt‖2

)

|v(i)|2 dt

)1/2

=(∑

i

|v(i)|2(∫ ti+1

ti

‖Ks‖2 ds +∫ ti+1

ti

‖Kt‖2 dt

))1/2

up to constant factors again. This tends to zero with the mesh size of the partition.Using the adaptation of operators, one sees easily that, once e(vi+1) has been

substituted with e(vi), it can be in turn substituted with e(v); the usual majorationsallow one to substitute it then with E(v). The convergence to zero of

∑

i

∫ ti+1

ti

∥∥∥∥

1

ti+1−ti

∫ ti+1

ti

K+s E(v) ds − K+

t E(v)

∥∥∥∥

2

dt

is then a simple consequence of the L2 martingale convergence theorem.

Now consider (3.9) in the case (ε, η) = (−, +): what we must show vanishes is∑

i

〈e(ui), (h−i k+

i − (H−K+)×i )e(vi)〉.

We show that

∑

i

⟨

e(ui),

(

h−i k+

i − ES

∫ ti+1

ti

H−t K+

t dt

)

e(vi)

⟩

vanishes. Its norm is easily shown to be smaller than

∑

i

∫ ti+1

ti

‖H−t ‖

∥∥∥∥

(1√

ti+1−tik+

i − K+t

)

e(vi)

∥∥∥∥ dt

�(∫

‖H−‖2)1/2

√√√√

∑

i

∫ ti+1

ti

∥∥∥∥

(1√

ti+1−tik+

i − K+t

)

e(vi)

∥∥∥∥

2

dt

and one concludes using Proposition 3.2.Last, consider (3.9) in the case (ε, η) = (◦, ◦): we prove that

⟨

e(u),∑

i

(h◦i k

◦i − (H ◦K◦)◦

i )a◦i e(v)

⟩

−→|S|→0

0.

This is equal to∑

i

u(i)v(i)〈e(ui), (h◦i k

◦i − (H ◦K◦)◦

i )e(vi)〉

154 YAN PAUTRAT

up to the usual last factor in the sum. The above line is equal to:

∑

i

u(i)v(i)

⟨

e(ui),1

ti+1−ti

∫ ti+1

ti

(H ◦t k◦

i − H ◦t K◦

t )e(vi) dt

⟩

=∫ ∞

0

u(i)v(i)

ti+1−ti

⟨

H ◦∗t e(ui),

1

ti+1−ti

∫ ti+1

ti

(K◦s − K◦

t )e(vi) ds

⟩

dt.

As in the proof of 3.1 we can replace (u(i)v(i))/(ti+1−ti) by u(t)v(t). The normof the integrated function is then smaller than

|u(t)v(t)|2sups‖H ◦∗s ‖sups‖K◦

s ‖‖E(u)‖‖E(v)‖which is integrable. By Lebesgue’s theorem, the considered quantity tends to zerowith the mesh size of the partition. �

3.2. A REMARK ON THE CLASSICAL ITÔ FORMULA

It is well known that the classical Itô formula for quantum stochastic integrals withrespect to any normal martingale is a consequence of the quantum Itô formula.Indeed, any normal martingale, that is, any martingale M with square bracket[M]t = t , can be identified with a multiplication operator on Fock space. Thatoperator has a quantum stochastic integral representation (see [At2]), so that itsangle bracket can be obtained from the Itô formula.

Therefore, we have proved that, once the normality of the martingale is known,the value of the angle bracket is deduced from the quantum stochastic integralrepresentation of the multiplication operator and the commutation relations forPauli matrices. There is nothing very deep here since the integral representationof the multiplication operator itself is derived from the structure equation of themartingale (see, for example, [At2]), but it may help and shed some light on thegeneral computations we have made.

The Brownian motion (Bt)t�0 can be identified to the operator process(a+

t + a−t )t�0. If we consider a partition S with constant steps δ, then the approxi-

mation of the multiplication operator by Bt will be∑

i|ti�t

√δ(a+

i +a−i ) plus some

terms which we have shown can be, in the limit when δ goes to zero, neglected.Besides, a+

i + a−i is σx so that

(√

δ(a+i + a−

i ))2δσ 2x = δI.

The operator δI is the approximation of the deterministic process (t)t�0. This, aswe have shown, implies that d〈B〉t = dt .

Another example is the compensated Poisson process (Nt − t)t�0 = (Xt)t�0. Itcan be identified with the operator process (a+

t +a−t +a◦

t )t�0. If we consider againa partition S with constant step δ, a+

t + a−t + a◦

t is projected to∑

i|ti�t (√

δa+i +

FROM PAULI MATRICES TO QUANTUM ITÔ FORMULA 155√

δa−i + a×

i ) plus asymptotically negligible terms. Since√

δa+i + √

δa−i + a×

i =√δσx − 1

2σz + 12I , we obtain

(√

δa+i + √

δa−i + a×

i )2 = δσ 2x + 1

4σ2z + 1

4I −− 1

2

√δ(σxσz + σzσx) + √

δσx − 12σz

= (√

δa+i + √

δa−i + a×

i ) + δI

because σxσz + σzσx = 0 and σ 2x = σ 2

z = I . That, as we have shown, implies thatd〈X〉t = Xt + t .

Much more interesting remarks can be made on the relation between randomwalks and normal martingales from the viewpoint of our approximation scheme,in particular in higher dimensional cases. In the paper [A-P], in collaboration withStéphane Attal, we show that limits of random walks can be completely determinedby the order of magnitude, with respect to the time scale, of the coefficients of their(discrete-time) structure equations.

References

[At1] Attal, S.: An algebra of noncommutative bounded semimartingales, square and anglequantum brackets, J. Funct. Anal. 124 (1994), 292–332.

[At2] Attal, S.: Classical and quantum stochastic calculus, In: Quantum Probability and RelatedTopics X, World Scientific, Singapore, 1998, pp. 1–52.

[At3] Attal, S.: Approximating the Fock space with the toy Fock space, In: Séminaire deProbabilités XXXVI, Springer, pp. 477–491.

[A-L] Attal, S. and Lindsay, J. M.: Quantum stochastic calculus with maximal operator domain,Ann. Probab. 32(1A) (2004), 488–529.

[A-M] Attal, S. and Meyer, P. A.: Interprétations probabilistes et extension des intégrales stochas-tiques non commutatives, In: Séminaire de Probabilités XXVII, Springer, 1993.

[A-P] Attal, S. and Pautrat, Y.: Some remarks on (n + 1)-level atom chains and n-dimensionalnoises, Ann. Inst. H. Poincaré, to appear.

[B-L] Belavkin, V. P. and Lindsay, J. M.: The kernel of a Fock space operator II, In: QuantumProbability and Related Topics VIII, World Scientific, Singapore 1993.

[Gui] Guichardet, A.: Symmetric Hilbert spaces and related topics, In: Lecture Notes in Math. 261,Springer, 1970.

[H-P] Hudson, R. L. and Parthasarathy, K. R.: Quantum Itô’s formula and stochastic evolutions,Comm. Math. Phys. 93(3) (1984).

[Mey] Meyer, P. A.: Quantum Probability for Probabilists, Lecture Notes in Math. 1538, Springer,New York, 1993.

[Pa1] Pautrat, Y.: Kernel and integral representations of operators on infinite dimensional toy Fockspace, In: Séminaire de Probabilités, to appear.

[Pa2] Pautrat, Y.: Des matrices de Pauli aux bruits quantiques, Thèse de doctorat de l’UniversitéJoseph Fourier, available on http://math-doc.ujf-grenoble.fr


Remarks on Radial Centres of Convex Bodies

I. HERBURT1, M. MOSZYNSKA2 and ZBIGNIEW PERADZYNSKI3

1Department of Mathematics and Information Sciences, Warsaw University of Technology,Pl. Politechniki 1, 00-601 Warszawa, Poland. e-mail: [email protected] of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland.e-mail: [email protected] of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2,02-097 Warszawa, Poland. e-mail: [email protected]

(Received: 9 June 2004; in final form: 25 February 2005)

Abstract. The paper concerns a family of selectors for convex bodies in Rn, radial centre maps,defined in the article of M. Moszynska, Looking for selectors of star bodies, Geom. Dedicata 81(2000), 131–147. A radial centre of a convex body A is the maximizer of a suitable generalized dualvolume of A. We give physical interpretations of the notion of radial centre and study its geometricproperties. We prove that these selectors are continuous with respect to the Hausdorff metric andsolve the problem of direct additivity for radial centre of order α, which corresponds to the dualvolume of order α.

Mathematics Subject Classifications (2005): 52A20, 52A40, 51P05, 85A25, 86A20.

Key words: convex body, continuous selector, directly additive selector, generalized dual volume,gravitational centre, radial centre.

0. Introduction

The notion of radial centre of a convex body associated with a function φ: R+ →R+ was introduced in [6] (see Section 1 for definition). In particular, if φ is theidentity, the (simple) radial centre of a convex body a in Rn is a unique pointr(A) ∈ A whose “mean distance from the boundary of A” is maximal.

There are various natural physical interpretations of this notion; they will bepresented in Section 1.

Section 2 concerns generalized dual volumes, the functions for which radialcentres are maximizers (compare [4]); they are proved to be strictly concave (seeTh. 2.1, a stronger version of the uniqueness theorem for radial centres).

Sections 3 and 4 deal with geometric properties of radial centres: in Section 3we prove continuity of radial centre maps with respect to the Hausdorff metric;in Section 4 we solve the problem of direct additivity of these selectors, raisedin [7]. Example 4.2, which proves that generally they are not directly additive, was

158 I. HERBURT ET AL.

checked by means of numerical methods by Katarzyna Piaskowska in her masterdissertation [8].

We follow, in principle, terminology and notation used in [9]. In particular, bd,int, cl, conv, lin, and aff are, respectively, boundary, interior, closure, convex hull,linear hull, and affine hull. For v ∈ Rn \ {0},

posv := {λv | λ > 0}.The k-dimensional Lebesgue measure in Euclidean k-dimensional space is λk, thek-dimensional spherical measure in a k-dimensional unit sphere is σk (or simply σ ,if it does not lead to a confusion). The k-dimensional Hausdorff measure is H k.

Bn and Sn−1 are the unit ball and the unit sphere in Rn, respectively. The classKn consists of compact convex subsets of Rn and Kn

0 is the class of convex bodies,i.e. members of Kn with nonempty interiors.�

Let F ⊂ Kn. A map s: F → Rn is called a selector for F if s(A) ∈ A forevery A ∈ F .

Let us recall that for every convex body A with 0 ∈ A the radial function�A: Sn−1 → R is defined by

�A(u) := sup{λ � 0 | λu ∈ A}.Evidently, if x ∈ A, then for every u ∈ Sn−1

�A−x(u) = ‖x − a‖,where a is the point of bdA ∩ (x + posu) most distant from x.

The radial sum A1+A2 of A1, A2 with 0 ∈ A1 ∩ A2 is defined by means of itsradial function:

�A1+A2:= �A1 + �A2 .

It is easy to see that

�A1+A2 � �A1+A2. (0.1)

1. Radial Centres and Their Physical Interpretations

Let us first recall the notion of radial centre of a convex body associated with somefunction φ: R+ → R+.

For A ∈ Kn0 , let �A: A → R be defined by the formula

�A(x) :=∫

Sn−1φ�A−x(u) dσ(u). (1.1)

� Schneider in [9] refers to any compact convex set as a convex body.

REMARKS ON RADIAL CENTRES OF CONVEX BODIES 159

It was proved that for n � 2 and φ concave and strictly increasing the function �A

has a unique maximizer, rφ(A), that is, a unique point at which this function attainsits maximum ([6], Th. 3.1); it is called the radial centre associated with φ.�

In particular, if n � 2 and φ is the identity, then �A(x) is (up to a constantfactor) the mean value of distance of x from bdA in the direction of u. Then theradial centre r(A): = rid(A) is referred to as simple radial centre of A. Thus thesimple radial centre of A is the maximizer of the function �A: A → R defined bythe formula

�A(x) :=∫

Sn−1�A−x(u) dσ(u).

Let now φ(t) = tα for some α ∈ R. If α ∈ (0; 1], then φ is concave and strictlyincreasing, whence, for n � 2, �A has a unique maximizer, r(α)(A), the radialcentre of order α of A.

Remark 1.1. Since the selector r(α) (for α ∈ (0; 1)) is equivariant under theisometries of Rn, that is, for every isometry f : Rn → Rn and A ∈ Kn

r(α)(f (A)) = f (r(α)(A)),

it follows that if A is symmetric with respect to an affine subspace E, thenr(α)(A) ∈ E.

For α = n a maximizer is not unique, because in this case, (1/n)�A(x) is equalto the volume of A, for every A ∈ Kn

0 and every x ∈ A (see [6], Example 3.2).For n = 1 every A is a closed segment in Rn: A = �(x0, x1) for some

x0, x1 ∈ R. It is easy to show that if φ is strictly (!) concave and monotone, then(x0 + x1)/2 is the unique maximizer of �A.

For α /∈ (0; 1] and α �= n � 2 some results on uniqueness were recentlyobtained by the first author [2].

We shall now consider some simple physical interpretations which can be asso-ciated with the notion of the radial centre of a convex body. Such interpretationscan be of some value also for the further development of the geometrical theory it-self. The physical intuition can suggest possible directions of research and possibleversions of theorems to prove. For example, it seems reasonable to investigate thenotion of radial centre also for non-convex bodies��, for which radial centre neednot be unique.

(A) The gravitational centre

A point mass M in R3 placed at a distance r from the reference point x is thesource of the gravitational potential at x, which is given by

(x) = −GM

r,

� In [6] it was denoted by Mφ(A). In Theorem 3.1 of [6] the assumption n � 2 was lacking.�� For star bodies it was introduced in [6].


where G is the gravitational constant. Hence, a uniformly distributed mass in aregion ⊂ R3 will create the gravitational potential at the point x according to thefollowing formula:

(x) = −G

∫

γ dλ3

r,

where r is the distance from the point x and γ is the mass density (here constant).Thus (x) is given by

(x) = −Gγ

2

∫S2

�2−x(u) du,

whence

(x) = −Gγ

2�(x)

for φ(t) := t2 (compare (1.1)).The minimum of the gravitational potential defines the place where the test

particle can stay at rest, i.e., a stable equilibrium. As it is visible from the lastformula, the gravitational centre does not coincide with the simple radial centre ofthe body, unless one considers a degenerate body, i.e. a body which is so thin inone dimension that with a reasonable approximation one can assume that the massis distributed over some plane domain.

(B) Gravitational centre of a thin two-dimensional body in R3

Let us consider a uniformly distributed matter over some plane domain ⊂R2 (embedded in R3). If now γ is the surface mass density in , then taking apoint x ∈ as the coordinate origin, one defines (in accordance with (A)) thegravitational potential at x by

(x) = −G

∫

γ

rdλ2 = −G

∫ 2π

0

∫ r(ψ)

0

γ

rr dr dψ = −Gγ

∫ 2π

0r(ψ) dψ,

where (ψ, r(ψ)) are polar coordinates of a boundary point of .Thus

(x) = −Gγ

∫S1

�−x(u) dσ(u) = −Gγ�(x)

for φ(t) = t , and the minimum of the gravitational potential is attained atthe equilibrium point of a test particle; this point corresponds to the simple radialcentre of the two-dimensional body .

(C) All the above results remain valid if instead of the gravitational interactionsone considers electric forces. Analogous formulae hold for the electric potentialcreated by a charge uniformly distributed in . Again, the test particle of the


opposite charge will stay at rest in the place where the so defined potential attainsits minimum. For a degenerate “thin body”, this equilibrium point is exactly thesimple radial centre of .

(D) Another interpretation, which can be generalized to higher dimensions, canbe provided by a transparent medium which is radiating the light (one may thinkof stars uniformly filling some region in the universe). In this case a source isradiating the light in such a way that its intensity at a distance r is proportional to1/r2. Assuming that the radiating sources are distributed uniformly in , the totallight intensity measured at the origin and coming from all points of the body isgiven by

I (0) ∼∫

dλ3

r2=

∫S2

∫ �(u)

0

r2

r2dr =

∫S2

�(u) dσ(u).

Thus, in this case, for a three-dimensional body in R3, the maximum of the intensityis measured at the simple radial centre of .

This result is valid for arbitrary dimension. The energy conservation law impliesthat in the n-dimensional Euclidean space the light intensity is decreasing with thedistance from the source, in such a way that it is inversely proportional to thedistance to the power n − 1. Therefore, if r is the distance from the coordinatecentre, then the intensity at the origin is given by

I (0) ∼∫

dλn

rn−1=

∫Sn−1

∫ �(u)

0

rn−1

rn−1dr =

∫Sn−1

�(u) dσ(u).

Hence, in this case the maximal intensity occurs in the simple radial centre of thebody.

2. Some Property of Generalized Dual Volumes for Convex Bodies

We shall now consider the function �A defined by (1.1) for A ∈ Kn0 . Let us note

that if φ(t) = tα for some α ∈ R, then �A(x) is the dual volume of A − x oforder α:

�A(x) = Vα(A − x)

(compare [1] (A.55) or [4]). Thus, it is natural to refer to the function �A, forany φ, as a generalized dual volume. We shall prove the following theorem, whichevidently is stronger than the uniqueness part of Theorem 3.1 in [6]. The idea of itsproof is similar to the idea of proof of the uniqueness.

THEOREM 2.1. Let n � 2. If A ∈ Kn0 and φ: R+ → R+ is concave and strictly

increasing, then the function �A defined by (1.1) is strictly concave.


Proof. Let x0, x1 ∈ A, x0 �= x1, t ∈ (0; 1), and let x := (1 − t)x0 + tx1.

Let Ai := A − xi for i = 0, 1. Then

A − x = (1 − t)A0 + tA1,

because A is convex.Hence, by (0.1) and by the assumptions on φ,

�A(x) � (1 − t)�A(x0) + t�A(x1). (2.2)

It remains to prove that the inequality in (2.2) is sharp.By dual Brunn–Minkowski inequality ([1] (B.28)), it follows that

Vn(A)1n = (1 − t)Vn(A0)

1/n + tVn(A1)1/n

� Vn((1 − t)A0+tA1)1/n,

(2.3)

and the equality holds if and only if there exists λ > 0 such that tA1 = λ(1− t)A0.Comparing the n-volumes, we obtain λ = t/(1−t), which is equivalent to A−x0 =A − x1; but this contradicts the assumption x0 �= x1.

Hence, by (2.3),

Vn(A − x) > Vn((1 − t)A0+A1),

that is,∫Sn−1

�nA−x >

∫Sn−1

((1 − t)�A0 + t�A1)n.

Thus, for some S ⊂ Sn−1 of a positive measure,

�A−x(u) > (1 − t)�A0(u) + t�A1(u) for u ∈ S,

whence for u ∈ S

φ�A−x(u) > (1 − t)φ�A−x0(u) + tφ�A−x1(u).

Integrating both sides over Sn−1, we obtain

�A((1 − t)x0 + tx1) > (1 − t)�A(x0) + t�A(x1),

i.e., �A is strictly concave. �

3. Continuity of Radial Centres

In [6] some extensions of the radial centre maps over a class of star bodies wereproved to be continuous with respect to the star metric.

We are now going to prove that radial centre maps for convex bodies are contin-uous with respect to the Hausdorff metric, which is topologically weaker than the


star metric (see [6], Th. 5.7�). Thus, of course, our result is stronger than Cor. 6.5in [6].

For every A ∈ Kn0 , let

U(A) := {u ∈ Sn−1 | ∃a, b b − a ∈ lin u and �(a, b) ⊂ bdA}.We start from the following.

LEMMA 3.1. Let Ak ∈ Kn0 and xk ∈ Ak for every k ∈ N ∪ {0}. If A0 = limH Ak

and x0 = lim xk, then for every u ∈ Sn−1 \ ⋃∞k=0 U(Ak)

�A−x0(u) = lim �Ak−xk(u).

Proof. Without any loss of generality we may assume that xk = x0 for every k.Indeed, let A′

k := Ak+x0−xk; then limH A′k = A = limH Ak and �A′

k−x0 = �Ak−xk.

Hence, let x0 ∈ A ∩ ⋂k�0 Ak, u ∈ Sn−1 \ ⋃

k�0 U(Ak), and

L+ := x0 + pos u, L− := x0 − pos u.

Let, further,

x+k ∈ L+ ∩ bdAk and x−

k ∈ L− ∩ bdAk

for every k ∈ N ∪ {0}.It suffices to prove that

x+0 = lim x+

k . (3.1)

For every k � 0 there are the following four possibilities:

(1)k x−k �= x0 �= x+

k ,(2)k x−

k �= x0 = x+k ,

(3)k x−k = x0 �= x+

k ,(4)k x−

k = x0 = x+k .

Passing to subsequences of (Ak)k∈N, we may assume that (x+k )k∈N is convergent

and exactly one of (1)k − (4)k is satisfied for all k � 1, i.e., for all members of thesequence (Ak)k∈N. Consequently, 16 conjunctions (i)k ∧ (j)0 for i, j ∈ {1, . . . , 4}are to be considered.

If L+ ∩ bdA0 is a singleton, i.e. one of the conditions (i)k ∧ (1)0, (i)k ∧ (2)0,and (i)k ∧ (4)0 is satisfied, then (3.1) holds. Thus the 16 conditions are reduced to(i)k ∧ (3)0 for i ∈ {1, . . . , 4}.

But, in view of Th. 14.3 in [3], by backward induction on the dimension of theaffine subspace, the above conjunction may hold only for i = 3, that means theonly possibility is:

x−k = x0 = x+

k and x−0 = x0 = x+

0 ,

which implies (3.1). �� The example at the end of proof of that theorem is wrong, but it is easy to find a correct one.


THEOREM 3.2. For every n � 2 and every concave and strictly increasing func-tion ϕ, the selector rϕ: Kn

0 → Rn is continuous with respect to the Hausdorffmetric.

Proof. Let A = limH Ak for a sequence (Ak)k∈N of convex bodies in Rn, andlet xk = rϕ(Ak) for every k. We may assume that (xk)k∈N is convergent. Let x =lim xk.

From Lemma 3.1 combined with the Lebegue majorized convergence theoremit follows that

�A(x) = lim �Ak(xk). (3.2)

We shall prove that x = rϕ(A). Suppose, to the contrary, that there exists y ∈ A

such that

�A(y) > �A(x). (3.3)

Since A = limH Ak, there exists (yk)k∈N such that yk ∈ Ak for every k andlim yk = y. Since �Ak

(xk) � �Ak(yk) for every k, applying again Lemma 3.1 for

the sequence (yk)k∈N we infer from (3.2) that �A(x) � �A(y), contrary to (3.3). �

4. Problem of Direct Additivity of Radial Centre Maps

In [7] various selectors for convex bodies were proved to be directly additive andthe problem was raised for radial centres.

For any Euclidean space E, let K(E) and K0(E) be, respectively, the class ofall compact convex subsets of E and the class of all convex bodies in E.

For any m � n, let Enm be the family of m-dimensional affine subspaces of Rn,

and let

En :=⋃m�n

Enm.

LEMMA 4.1 (see [7]). Let E ∈ Enm for some m � n and let f1, f2: E → Rm be

isometries. If A ∈ K0(E) and a selector s: Km0 → Rm is equivariant under the

isometries of Rm, then

f −11 (s(f1(A))) = f −1

2 (s(f2(A))).

Let now ⊕ be the direct sum: if Rn = E1 ⊕ E2 for two affine subspaces E1, E2

of Rn with positive dimensions, then for every Xi ⊂ Ei , i = 1, 2,

X1 ⊕ X2 := X1 + X2.

Thus, direct sum operation is a restriction of the Minkowski addition.


In view of Lemma 4.1, for a given n, every sequence (s(m): Km → Rm)m�n ofequivariant selectors determines the family of selectors,

s := (sE: K(E) → E)E∈En,

defined as follows: for every E ∈ Enm and A ∈ K(E),

sE(A) := f −1s(m)f (A), (4.1)

where f : E → Rm is an isometry.Let now s = (sE)E∈En be the family determined by a sequence (s(m))m�n of

equivariant selectors (see (4.1)) and let s := s(n). The family s is directly additive(or, the sequence (s(m))m�n is directly additive) if and only if

s(A1 ⊕ A2) = sE1(A1) + sE2(A2) (4.2)

whenever Ei = affAi and Rn = E1 ⊕ E2 with E1, E2 orthogonal.We shall now consider the radial centre map r(α) of order α for α ∈ (0; 1) (see

Section 1). As was already mentioned in Section 2, r(α)(A) is the point of A atwhich Vα(A − x) attains its maximum.

Generally, the family r(α) for α ∈ (0; 1) is not directly additive. The followingexample concerns the case n = 3 and α = 1/2.�

EXAMPLE 4.2. Let (e1, e2, e3) be the canonical basis in R3. Let

E1 = lin(e1, e2), E2 = line3,

A1 = �(0, e1, e2) ⊂ E1, A2 = �(se3, −se3), and A = A1 ⊕ A2.

Since L := {(t, t, 0) | t ∈ R} is the symmetry line of the triangle A1 and 0 isthe midpoint of the segment A2, in view of Remark 1.1,

r(1/2)

E1(A1) ∈ L and r

(1/2)

E2(A2) = 0.

Similarly, A is symmetric with respect to L, whence r(1/2)(A) ∈ L.

We are going to show that

r(1/2)(A) �= r(1/2)

E1(A1) + r

(1/2)

E2(A2),

that is, r(1/2)(A) �= r(1/2)

E1(A1).

We first calculate �A1(x) for x = (t, t, 0) and t ∈ [0; 1/2].Let

β1(t) := π

4, β2(t) := arctan

1 − t

t, β3(t) := arctan

1

1 − 2t,

� In [8] also the case n = 3 and α = 1 (the simple radial centre) was considered.


Figure 1. Graph of the function t �→ 2f (t) = �A1(t, t, 0).

h1(t) := t, and h2(t) :=√

2

2(1 − 2t).

Then, as is easy to check, �A1(x) = 2f (t), where

f (t) := √h1(t)

∫ β1(t)

0

1√cos θ

dθ +

+ √h1(t)

∫ β2(t)

0

1√cos θ

dθ + √h2(t)

∫ β3(t)

0

1√cos θ

dθ

(see Figure 1).Let us now calculate �A(x) for x = (t, t, 0) and t ∈ [0; 1/2]. Evidently, since

A depends on the parameter s, so does �A(x) and its maximizer (see Figure 2).Let F1, . . . , F5 be the facets of A:

F1 := �(0, e1) ⊕ A2, F2 := �(0, e2) ⊕ A2, F3 := �(e1, e2) ⊕ A2,

F4 := A1 ⊕ {−se3}, and F5 := A1 ⊕ {se3}.Let

Sj := {u ∈ S2 | (x + pos u) ∩ Fj �= ∅},


Figure 2. Graphs of functions t �→ �A(t, t, 0) for different values of s compared with thegraph of t �→ �A1(t, t, 0).

whence

S2 =5⋃

j=1

Sj .

If

Ij :=∫

Sj

�αA−x(u) dσ(u),

then

�A(x) =5∑

j=1

Ij (t, s) = 2I1(t, s) + I3(t, s) + 2I5(t, s).

Since �A−x(u) is equal to the length of segment with endpoints x and (x +pos u) ∩ bdA, it follows that, choosing suitable spherical coordinates for each ofS1, S3, S5, we can calculate I1(t, s), I3(t, s), and I5(t, s):

Let

ψ(d, θ, a) := arctan

(a

d cos θ

)


and

g(d, θ0, a) :=∫ θ0

0

∫ ψ(d,θ,a)

0

(d

cos ψ

)α

sin ψ dψ dθ.

Then

I1(t, s) = 2g

(t, arctan

s

t, t

)+ 2g

(t, arctan

t

s, s

)+

+ 2g

(t, arctan

s

1 − t, 1 − t

)+ 2g

(t, arctan

1 − t

s, s

),

I3(t, s) = 4g

((1 − 2t)

√2

2, arctan(

√2s),

√2

2

)+

+ 4g

((1 − 2t)

√2

2,π

2− arctan(

√2s), s

),

and

I5(t, s) = 2

(g

(s,

π

4, t

)+ g

(s, arctan

1

1 − 2t,

√2

2(1 − 2t)

)).

The values of the functions t �→ �A1(t, t, 0) and t �→ �A(t, t, 0) and theirmaximizers were approximately found by means of numerical methods (see Fig-ure 3).

The maximizers were obtained by method of finding zero of derivative. New-ton’s algorithm using the secant method of approximation of derivative was ap-plied. Computations were performed within double precision arithmetic, i.e., with16 decimal digits of the mantissa (see [8]).

We shall now prove that radial centre of order α (for α ∈ (0; 1)) satisfiescondition (4.2) for some particular pairs (A1, A2).

We begin with two lemmas.

LEMMA 4.3. Let Rn = E1 ⊕ E2 with E1, E2 orthogonal and dim Ei = ni fori = 1, 2. Let Ai ∈ K(Ei), A = A1 ⊕ A2, and let Bni be the unit ball in Ei . Ifφ(t) = tα for α ∈ (0; 1), then, for every x ∈ A,

�A(x) =∫

bd(Bn1⊕Bn2 )

1

‖v‖n�α

A−x

(v

‖v‖)

dv.

Proof. By [9] (4.2.25) for every K ∈ Kn0 with 0 ∈ intK and for every Hn−1-

measurable X ⊂ bdK ,

Hn−1(X) =∫

π(X)

�K(u)n−1

u ◦ n(K, u)dσ(u),


Figure 3. Distance of maximizers of functions t �→ �A1(t, t, 0) and t �→ �A(t, t, 0) fordifferent values of s.

where n(K, u) is an outer normal unit vector at pos u ∩ bdK (which is uniqueσ -almost everywhere) and π : bdK → Sn−1 is the central projection:

π(v) := v

‖v‖ .

Thus,∫X

�αA−x(v/‖v‖)

‖v‖ndHn−1(v) =

∫π(X)

�αA−x(u)

‖π−1(u)‖n· �K(u)n−1

u ◦ n(K, u)dσ(u). (4.3)

Let now K := Bn1 ⊕ Bn2 and X := bdK . Then

X = Sn1−1 ⊕ Bn2 ∪ Bn1 ⊕ Sn2−1.

If u = v/‖v‖ for some v ∈ X, then v = v1 + v2, where either v1 ∈ Sn1−1 andv2 ∈ Bn2 or v1 ∈ Bn1 and v2 ∈ Sn2−1. We may assume that v /∈ Sn1−1 ⊕ Sn2−1.Then

n(K, u) ={

v1 if v ∈ Sn1−1 ⊕ intBn2 ,v2 if v ∈ intBn1 ⊕ Sn2−1.

Hence,

u ◦ n(K, u) = 1

‖π−1(u)‖


for almost every u ∈ Sn−1; therefore, the right-hand side of (4.3) is equal to �A(x),because it is equal to

∫Sn−1

�αA−x(u)

‖π−1(u)‖n· ‖π−1(u)‖n−1 · ‖π−1(u)‖ dσ(u).

Consequently, by (4.3),

�A(x) =∫

bd(Bn1⊕Bn2 )

�αA−x(v/‖v‖)

‖v‖ndHn−1(v). �

Let, as above, n = n1 + n2 and let the unit balls Bn1, Bn2 lie in orthogonalsubspaces E1, E2 with dim Ei = ni . Define

Z1 := Sn1−1 ⊕ Bn2 and Z2 := Bn1 ⊕ Sn2−1;thus bd(Bn1 ⊕ Bn2) = Z1 ∪ Z2.

LEMMA 4.4. Let Rn = E1 ⊕ E2 with E1, E2 orthogonal and dim Ei = ni fori = 1, 2. Let Ai ∈ K(Ei), A = A1 ⊕ A2, and let Bni be the unit ball in Ei . Forx = x1 + x2 with xi ∈ Ai , let

C1(x) := {v ∈ Z1 | pos v ∩ (bd(A1 − x1) ⊕ (A2 − x2)) �= ∅},

D1(x) := {v ∈ Z1 | pos v ∩ ((A1 − x1) ⊕ bd(A2 − x2)) �= ∅},

C2(x) := {v ∈ Z2 | pos v ∩ ((A1 − x1) ⊕ bd(A2 − x2)) �= ∅},

D2(x) := {v ∈ Z2 | pos v ∩ (bd(A1 − x1) ⊕ (A2 − x2)) �= ∅}.

Then

Zi = Ci(x) ∪ Di(x)

and

�A(x) =∫

C1(x)∪D2(x)

�αA1−x1

(v1)‖v‖1−n dv +

+∫

C2(x)∪D1(x)

�αA2−x2

(v2)‖v‖1−n dv, (4.4)

where v = v1 + v2 with vi ∈ Bni for i = 1, 2.

Proof. Let v ∈ bd(Bn1 ⊕ Bn2) and v = v1 + v2 for vi ∈ Bni . If v ∈ C1(x), then

�A−x

(v

‖v‖)

=√

�2A1−x1

(v1) + w22,


where ‖w2‖ = ‖v2‖�A1−x1(v1); thus

�A−x

(v

‖v‖)

= �A1−x1(v1)‖v‖.

If v ∈ D1(x), then

�A−x

(v

‖v‖)

=√

‖w1‖2 + �2A2−x2

(v2),

where ‖w1‖ = ‖v1‖�A2−x2(v2); thus

�A−x

(v

‖v‖)

= �A2−x2(v2)‖v‖.

Analogously, if v ∈ C2(x), then

�A−x

(v

‖v‖)

= �A2−x2(v2)‖v‖,

and if v ∈ D2(x), then

�A−x

(v

‖v‖)

= �A1−x1(v1)‖v‖.

Hence, by Lemma 4.3, we obtain (4.4). �We shall now consider a particular situation and prove condition (4.2) for the

radial centre of order α in this particular case.

THEOREM 4.5. Let n = 2m for some natural m � 2 and let Rn = E1 ⊕ E2 withdim E1 = m = dim E2. If Ai ∈ K0(Ei) for i = 1, 2, and there exists an isometryf : E1 → E2 such that f (A1) = A2, then for every α ∈ (0; 1]

(r(α))(A1 ⊕ A2) = r(α)E1

(A1) + r(α)E2

(A2).

Proof. For every xi ∈ Ai, i = 1, 2 and for every u ∈ Sm−1,

�A1−x1(u) = �A2−x2(f (u)).

Applying formula (4.4) and the Fubini Theorem, we infer that

�A(x) =∫

Sm−1⊕Bm

�αA1−x1

(v1)‖v‖α−n dv +∫

Bm⊕Sm−1�α

A2−x2(v2)‖v‖α−n dv

=∫

Sm−1�α

A1−x1(v1) dσm−1(v1)

∫Bm

dλm(v2)

(√

1 + ‖v2‖2)n−α+

+∫

Sm−1�α

A2−x2(v2) dσm−1(v2)

∫Bm

dλm(v1)

(√

1 + ‖v1‖2)n−α.


Thus, setting

β :=∫

Bm

dλm(v)

(√

1 + ‖v‖2)n−α,

we obtain

�A(x) = β(�A1(x1) + �A2(x2)).

Since, evidently, x1 is the maximizer of �A1 if and only if f (x1) is the maximizerof �A2 , it follows that x is the maximizer of �A if and only if x = x1 +f (x1). Thiscompletes the proof. �

Acknowledgement

The authors are grateful to Katarzyna Piaskowska for her contribution.

References

1. Gardner, R.: Geometric Tomography, Cambridge University Press, Cambridge, 1995.2. Herburt, I.: Extremizers of dual volumes for convex and star bodies, in preparation.3. Leichtweiss, K.: Konvexe Mengen, Springer, Berlin, 1980.4. Lutwak, E.: Intersection bodies and dual mixed volumes, Advances in Math. 71(2) (1988), 232–

261.5. Moszynska, M.: Remarks on the minimal rings of convex bodies, Studia Sci. Math. Hung. 35

(1999), 1–20.6. Moszynska, M.: Looking for selectors of star bodies, Geom. Dedicata 81 (2000), 131–147.7. Moszynska, M.: On directly additive selectors for convex and star bodies, Glasnik Mat. 39(59)

(2004), 145–154.8. Piaskowska, K.: Numerical localization of values of some selectors for convex bodies (in Polish),

Master Dissertation (unpublished), Dept. of Math., Warsaw University, 2004.9. Schneider, R. Convex Bodies: the Brunn–Minkowski Theory, Cambridge University Press,

Cambridge, 1993.


Ergodic Properties of the Quantum GeodesicFlow on Tori

SŁAWOMIR KLIMEK1 and WITOLD KONDRACKI2

1Department of Mathematics, Indiana University Purdue University Indianapolis,402 N. Blackford St., Indianapolis, IN 46202, USA2Institute of Mathematics, Polish Academy of Sciences, Ul. Sniadeckich 8, Warsaw, Poland

(Received: 21 January 2004; in final form: 23 February 2005)

Abstract. We study ergodic averages for a class of pseudodifferential operators on the flatN -dimensional torus with respect to the Schrödinger evolution. The later can be consider a quan-tization of the geodesic flow on T

N . We prove that, up to semi-classically negligible corrections,such ergodic averages are translationally invariant operators.

Mathematics Subject Classifications (2000): 58J50, 58J40, 81S10.

Key words: ergodic averages, geodesic flow, quantization, quantum chaos.

1. Introduction

In this paper we study time averages for a class of pseudodifferential operators onthe N -dimensional torus T

N = RN/(2πZ)N equipped with the natural flat Rie-

mannian metric. More generally, instead of the torus one can consider a compactmanifold X and a Riemannian metric g on X. Denote by � the correspondingLaplace operator (on functions).

Let Q be a quantization of the cotangent space T ∗X of X, i.e. a linear cor-respondence F → Q(F), that to a class of functions F : T ∗X → C associatesoperators Q(F) in L2(X, dµ). Let A be an algebra generated by all the operatorsQ(F). The most popular choices for Q(F) are pseudo-differential operators on X

with symbols F .The averages are taken with respect to the time evolution given by a unitary

group {eitH } of operators in L2(X, dµ}. There are two common choices for thegenerator H of the evolution: H = √−� or H = − 1

2�. The later is the choice inthis paper. This operator has discrete spectrum, H ·φn = µnφn. We can arrange µn

so that µ1 � µ2 · · · → ∞.The ergodic averages are, by definition, the following expressions:

〈Q(F)〉 := limT →∞

1

T

∫ T

0e−itHQ(F) eitH dt. (1)

174 SłAWOMIR KLIMEK AND WITOLD KONDRACKI

The problem considered in this paper is to compute such averages as an opera-tion on complete symbols and in particular to relate them to the geodesic flow onT ∗X. Zelditch studied such averages and the corresponding symbol operation onhyperbolic surfaces in [14].

This problem belongs to the area of quantum chaos, which studies semi-classicalasymptotics of eigenfunctions and eigenvalues of quantum systems. Here H is thequantum hamiltonian of the free particle on X, and can be considered a quantiza-tion of the geodesic flow on T ∗X. Detailed survey of this area is contained in [12]covering both the motivation and the overview of the results.

One of the main problems in that theory is to study possible limit points of thesequence of functionals

{F → (φn, Q(F)φn)}. (2)

The point is that the ergodic averages (1) may be more tractable than generaloperators while not changing the limit structure of (2) since:

(φn, Q(F)φn) = (φn, 〈Q(F)〉φn),

which gives a motivation for our study.There are numerous results and conjectures concerning those limit points which

depend on whether the corresponding geodesic flow is ergodic or not. The case ofergodic geodesic flow is more interesting and will be briefly reviewed in Section 2.If the geodesic flow is not ergodic but completely integrable, as is the case for T

N

considered in this paper, the limits of (2) are complicated, possibly reflecting dif-ferent ergodic components (invariant tori) of the geodesic flow. Some results in thatdirection are described in [11] and [3]. The main outcome of our analysis is thatthe ergodic averages are, up to semiclassically negligible correction, equal to theirclassical averages:

〈Q(F)〉 = Q(F ) + aF ,

where F (p) = 1(2π)N

∫TN F (x, p) dNx, and aF is a semi-classically negligible

operator – see below. Moreover, if N = 1, then aF is a compact operator.The paper is organized as follows. In Section 2 we shortly discuss ergodic av-

erages in different situations. Section 3 describes the class of pseudodifferentialoperators used for our analysis. In Section 4 we study the ergodic averages. Thatsection contains the main result.

2. Semi-classical Ergodicity

In what follows we give a short overview of quantum ergodic theorems. Let A bean algebra of operators acting on a Hilbert space H , and let ρt be a one-parametergroup of ∗-automorphisms of A of the form

ρt(a) = eitH a e−itH , a ∈ A,

ERGODIC PROPERTIES OF THE QUANTUM GEODESIC FLOW ON TORI 175

where H is a self-adjoint, usually unbounded operator on H . The main object ofergodic theory are time averages

〈a〉 := limT →∞

1

T

∫ T

0ρt(a) dt,

where the limit above is taken with respect to a suitable topology. Under convenientassumptions, this limit exists and is ρt -invariant. The dynamical system (A, ρt ) isthen called ergodic if the following statement holds (ergodic theorem):

〈a〉 = τ(a)I,

where τ is a ρt -invariant state on A. For examples of this situation see [5–7].In many important quantum mechanical problems this scenario does not apply

[4, 12]. Let us assume that the spectrum of H is purely discrete:

Hφn = µnφn,

and µn → ∞. One considers the following special, invariant state on A:

τ(a) = limE→∞

τE(a) := limE→∞

1

#{µn � E}∑

µn�E

(φn, aφn). (3)

Intuitively this state captures the information from very large eigenvalues of H

(= high energies) and is usually taken as the starting point of the semi-classicalergodic theory. An operator a is called semi-classically negligible if

τ(a∗a) := limE→∞

τE(a∗a) = 0.

The dynamical system (A, ρt ) is then called semiclassically ergodic if the follow-ing statement holds (semiclassical ergodic theorem):

〈a〉 = τ(a)I + Ca, (4)

where Ca is semi-classically negligible. A theorem of Zelditch [10] gives sufficientconditions for (4) to hold. Let πτ be the GNS representation of A associated with τ ,where τ is given by (3). The theorem says that if we assume that the algebra πτ (A)

is commutative, and that πτ (ρt ) is ergodic (in the classical sense), then the system(A, ρt , τ ) is semi-classically ergodic i.e. (4) holds.

We now describe the main example of a semi-classically ergodic system. LetQ(f ) be a ψDO of order zero on X with symbol f : T ∗X �→ C. Let fP be theprincipal symbol of Q(f ). It is a homogeneous function on the cotangent bundleso, effectively, it is a function on the unit sphere bundle of the cotangent bundlefP : ST∗X �→ C. Let γt be the geodesic flow on ST∗X and denote by dL theLiouville measure on ST∗X. This measure is invariant with respect to the geodesicflow γt . Moreover, if (X, g) has negative curvature then γt is ergodic.


Take H := √−�. Using ψDO theory one shows that

τ(Q(f )) := limE→∞

1

#{µn � E}∑

µn�E

(φn, Q(f )φn) =∫

ST∗XfP dL.

Let A be the C∗-algebra obtained as the norm closure of the algebra of order zero

ψDO on X. It is well known, and not difficult to prove, that A has the followingstructure:

0 −→ K −→ Aσ−→ C(ST∗X) −→ 0.

Here K is the ideal of compact operators in L2(X, dµ) and σ is called the symbolmap. It follows that if πτ the GNS representation of A associated with τ , wehave πτ (A) = C(ST∗X), which is abelian. The Egorov’s theorem implies nowthat πτ (ρt ) = γt , the geodesic flow on ST∗X. So, by Anosov’s results [1], if X

has negative (sectional) curvature, the ergodic theorem (4) follows. From here itis not difficult to see that the following theorem, due to Zelditch [13], Colin deVerdiere [2] and Schnirelman [9], is true: if X has negative curvature, there is asubset S ∈ N of density 1 such that

limn∈S,n→∞(φn, Q(f )φn) =

∫ST∗X

fP dL. (5)

It was conjectured in [8] that the remainder term Ca in (4) is compact so that thelimit in (5) exists for every subsequence. If true, this would imply that no “scars”exist for such systems.

Finally, when X = TN , the geodesic flow is not ergodic but is completely

integrable and the corresponding quantum system is not semiclassically ergodic.The main result of this paper establishes an analog of (4) in this case.

3. Operators

First we introduce a class of pseudodifferential operators suitable for our analysis.In what follows T

N is the N -dimensional torus considered as the quotient TN =

RN/(2πZ)N , and equipped with the induced flat Riemannian metric. The cotangent

space T ∗(TN) of TN is naturally identified with T

N × RN . The operators we are

interested in are constructed using symbols, which are functions on TN × R

N .To specify the class of functions we need the following notation: for a boundedfunction F on T

N ×RN , continuous in the T

N direction, let F be its partial Fouriertransform along T

N i.e.

F (k, p) = 1

(2π)N

∫TN

F (x, p) e−ikx dNx.

Define Cr(TN ×R

N), r > N , to be the space of functions F on TN ×R

N such that

‖F‖r := sup(k,p)∈ZN×ZN

|(1 + |k|2)r/2F (k, p)| < ∞. (6)


It should be noted that in the above definition only the values of F(x, p) on T×ZN

matter as justified below by the formula (8) for the operator Q(F) defined by F .Also Cr(T

N × RN) is a Banach space and linear combinations of functions of the

form f (x)g(p), where f is a trigonometric polynomial, form a dense subspace inCr(T

N × RN). Moreover we have:

PROPOSITION 1.

1

Cr

sup(x,p)∈TN×ZN

|F(x, p)| � ‖F‖r � sup(x,p)∈TN×ZN

|(1 − �)r/2F(x, p)|

where � is the laplacian on TN and

Cr =∑k∈ZN

(1 + |k|2)−r/2. (7)

(As stated above we assume r > N throughout the paper.)Proof. We have:

sup(x,p)∈TN×ZN

|F(x, p)| = sup(x,p)∈ZN×ZN

∣∣∣∣∑k∈ZN

F (k, p) eikx

∣∣∣∣� sup

p∈ZN

∑k∈ZN

|F (k, p)|

� sup(k,p)∈ZN×ZN

|(1 + |k|2)r/2F (k, p)| ×

×∑k∈ZN

(1 + |k|2)−r/2,

which proves the first inequality. On the other hand we have:

sup(k,p)∈ZN×ZN

|(1 + |k|2)r/2F (k, p)| = sup(k,p)∈ZN×ZN

| (1 − �)r/2F(k, p)|

� sup(x,p)∈TN×ZN

|(1 − �)r/2F(x, p)|,

because f (k) � supx∈TN |f (x)|. �To each symbol F ∈ Cr(T

N × RN) we associate a bounded operator Q(F) in

L2(TN, dNx) by the following, global version of the usual formula:

Q(F)ψ(x) = 1

(2π)N

∑k∈ZN

∫TN

F (x, k) eik(x−y)ψ(y) dNy.

Notice also that writing ψ ∈ L2(TN, dNx) in the Fourier series:

ψ(x) =∑k∈ZN

ψk eikx,


we have

Q(F)ψ(x) =∑k∈ZN

ψkF (x, k) eikx. (8)

LEMMA 2. We have

‖Q(F)‖ � Cr‖F‖r ,

where Cr is given by (7).Proof. Note that in the formulas below the summation indexes run over Z

N . Westudy (ψ, Q(F)ψ) with ψ(x) = ∑

ψk eikx . Using (8) we obtain:

(ψ, Q(F)ψ)=∑k,l,m

ψkψmF (l, m)

∫TN

ei(l+m−k)x dNx

(2π)N

=∑k,m

ψkψmF (k − m, m)

=∑k,m

ψkψmF (k − m, m)(1 + (k − m)2)r/2

(1 + (k − m)2)r/2.

It follows that:

|(ψ, Q(F)ψ)| = ‖F‖r

∑k,m

|ψk| |ψm|(1 + (k − m)2)−r/2

= ‖F‖r

∑k,l,m

|ψk| |ψm|(1 + l2)−r/2∫

TN

ei(l+m−k)x dNx

(2π)N.

Let ψ(x) = ∑k |ψk| eikx . Notice that ‖ψ‖ = ‖ψ‖. We can use ψ to write the

above estimate as

|(ψ, Q(F)ψ)|� ‖F‖r

(ψ,

∑l

(1 + l2)−r/2 eilxψ

)

� ‖F‖r supx∈TN

∣∣∣∣∑

l

(1 + l2)−r/2 eilx

∣∣∣∣‖ψ‖2 � Cr‖F‖r‖ψ‖2,

and the claim follows. �PROPOSITION 3. If F ∈ Cr(T

N × RN) and |F(x, p)| → 0 as p → ∞, then

Q(F) is a compact operator.Proof. First notice that if |F(x, p)| → 0 as p → ∞ then also |F (k, p)| → 0 as

p → ∞. Indeed:

|F (k, p)| =∣∣∣∣ 1

(2π)N

∫TN

F (x, p) e−ikx dNx

∣∣∣∣ � supx∈TN

|F(x, p)| → 0.


Consider now FL(x, p) := ∑|k|�L F (k, p) eikx . It follows that |FL(x, p)| → 0

as p → ∞. Additionally FL → F in Cr(TN × R

N), which by Lemma 2 impliesQ(FL) → Q(F) in norm. Finally Q(FL) is a compact operator. Indeed, for eachFourier coefficient F (k, p) we have

Q(F (k, ·)) einx = F(k, n) einx.

This means that Q(F (k, ·)) has discrete spectrum equal to F(k, n), n ∈ ZN which

goes to 0 as n → ∞, implying that Q(F (k, ·) and also Q(FL) are compactoperators. Finally, Q(F) is compact as a norm limit of compact operators. �

The following observation is worth a notice even though it is not used in theremainder of the paper. It gives more information on the topological properties ofthe quantization map F → Q(F).

PROPOSITION 4. If Fn → F pointwise (a.e.) and if {Fn} have uniformly boundednorms (in Cr(T

N × RN)) then Q(Tn) → Q(T ) weakly.

Proof. First notice that Proposition 1 and uniform boundedness of Cr(TN ×R

N)

norms implies that {Fn} are uniformly bounded, that is, their sup norms are uni-formly bounded. It follows using Lebesque dominated convergence theorem thatFn → F pointwise. To show weak convergence of Q(Fn) we study (ψ, Q(Fn)ψ)

just as in Lemma 2. We have

(ψ, Q(Fn)ψ) =∑k,m

ψkψmFn(k − m, m),

and by the previous remark the inside of the sum converges pointwise toψkψmF (k − m, m) as n → ∞. Also

|ψkψmFn(k − m, m)| =∣∣∣∣ψkψmFn(k − m, m)

(1 + (k − m)2)r/2

(1 + (k − m)2)r/2

∣∣∣∣�

(sup

n

‖Fn‖r

)|ψk||ψm|(1 + (k − m)2)−r/2.

The last expression is summable just as in the proof of Lemma 2, so using Lebesquedominated convergence theorem again concludes the proof. �

4. Ergodic Averages

Let H be the following operator in L2(TN, dNx):

H := −1

2� = −1

2

(∂2

∂x1+ · · · + ∂2

∂xN

),


and let φn = einx ∈ L2(TN, dNx). Then Hφn = µnφn with µn = n2/2. We areinterested in studying the norm limits:

〈Q(F)〉 := limT →∞

〈Q(F)〉T := limT →∞

1

T

∫ T

0e−itHQ(F) eitH dt,

called ergodic averages.The following general lemma deals with ergodic averages of compact operators.

LEMMA 5. If C is a compact operator and H has discrete spectrum with finitemultiplicities, then limT →∞ 1

T

∫ N

0 e−itHC eitH dt exists in norm and is a compactoperator.

Proof. We study the weak limit first:

〈C〉 = w − limT →∞〈C〉T = w − lim

T →∞1

T

∫ T

0e−itHC eitH dt.

The matrix elements with respect to the basis of eigenvectors of H are

(φi, 〈C〉T φj ) = (φi, Cφj )1

T

∫ T

0eit (λi−λj ) dt.

Consequently,

(φi, 〈C〉φj ) ={

0, if λi = λj ,

(φi, Cφj ), otherwise.(9)

We need the following notation. Writing the spectral decomposition of the Hilbertspace H = ⊕iHi with respect to operator H , denote by Pi the orthogonal pro-jection onto Hi , the eigenspace corresponding to the eigenvalue λi . Projections Pi

are mutually orthogonal PiPj = δijPi . We will also need P�n := ∑i�n Pi , and

P�n := ∑i�n Pi . We have

PiP�n = Pi,

if i � n. In this notation we have

〈C〉 =∑

i

PiCPi,

where the limit defining the series is in the weak topology. We claim that 〈C〉 iscompact. To prove it we show that

∥∥∥∥∑i�n

PiCPi

∥∥∥∥ → 0 as n → ∞.


This implies that 〈C〉 is compact as a norm limit of compact operators. We com-pute:

∥∥∥∥∑i�n

PiCPix

∥∥∥∥2

=∑i�n

‖PiCPix‖2 =∑i�n

‖PiP�nCPix‖2

� ‖P�nC‖2∑i�n

‖Pix‖2 � ‖P�nC‖2‖x‖2.

Consequently,∥∥∥∥

∑i�n

PiCPi

∥∥∥∥ � ‖P�nC‖ → 0 as n → +∞,

since C is compact.So far we have

w − limT →∞

〈C〉T = 〈C〉,

with compact 〈C〉. Observe also that the averaging is a contraction:

‖〈C〉T ‖ � ‖C‖.To prove that 〈C〉T converges to 〈C〉 in norm we set up an approximation argument.To this end we introduce Cn := P�nCP�n. We have:

• Cn → C in norm (by compactness of C).• 〈Cn〉T → 〈C〉n := P�n〈C〉P�n weakly, but since Cn are finite dimensional, it

means in norm.• 〈C〉n → 〈C〉 in norm (by compactness of 〈C〉).

It follows that in the estimate

‖〈C〉T − 〈C〉‖ � ‖〈C − Cn〉T ‖ + ‖〈Cn〉T − 〈C〉n‖ + ‖〈C〉n − 〈C〉‖,all three terms are small. �COROLLARY 6. F ∈ Cr(T

N ×RN) and |F(x, p)| → 0 as p → ∞, then 〈Q(F)〉

is a compact operator.Proof. This immediately follows from Proposition 3 and Lemma 5. �Next we study ergodic averages of arbitrary operators Q(F). To this end we

obtain an explicit formula for eitHQ(F) e−itH .

LEMMA 7. If F ∈ Cr(T × RN) then

eitHQ(F) e−itH = Q(Ft), (10)


where Ft ∈ Cr(TN × R

N) and

Ft(x, p) = e−it�x/2(F (x + tp, p)), (11)

where �x = ∂2

∂x1+· · ·+ ∂2

∂xNis the Laplace operator in the x-coordinate. In Fourier

transform:

Ft (k, p) = eit (k2/2+kp)F (k, p). (12)

Notice that in (11) the argument shift x → x + tp is precisely the geodesic flowon the cotangent space of T

N with respect to the flat metric. The action of e−it�r/2

on F in (11) is a quantum effect, and, as we will see, it does influence the ergodicproperties of the operators.

Proof. Since F and Ft differ by a phase it follows that F and Ft have the samenorm:

‖Ft‖r = ‖F‖r .

Because of this, the usual 2ε argument shows that it is enough to verify (10) for adense set of F ’s. Indeed, if Fn → F and (10) holds for Fn then in the followingestimate all terms are small:

‖eitHQ(F) e−itH − Q(Ft)‖ � ‖eitH (Q(F) − Q(Fn))e−itH‖ +

+ ‖eitHQ(Fn)e−itH − Q((Fn)t )‖ +

+ ‖Q((Fn)t ) − Q(Ft)‖.Also, both sides of (10) are linear in F so it is enough to verify the hypothesis forfunctions in the following form: F(x, p) = eikxg(p), since linear combinationsof them form a dense set in Cr(T

N × RN). Applying eitHQ(F)e−itH to ψ(x) =∑

l ψl eilx we obtain

eitHQ(F) e−itHψ(x) =∑

l

eit (k+l)2/2 eikxg(l) e−it l2/2ψl eilx

= eitk2/2∑

l

eik(x+t l)g(l)ψl eilx

= Q(eitk2/2 eik(x+tp)g(p))ψ(x) = Q(Ft)ψ(x),

which concludes the proof. �Now we use Lemma 7 to explicitly calculate ergodic averages of Q(F). It turns

out that long time averaging simply drops most frequencies out from F as describedby the following lemma.

LEMMA 8. The norm limit 〈Q(F)〉 := limT →∞ 1T

∫ T

0 e−itHQ(F) eitH dt existsand 〈Q(F)〉 = Q(〈F 〉), where

〈F 〉 =∑

{k∈ZN : kp+k2/2=0}F (k, p) eikx . (13)


Proof. The following observations:

• 〈·〉 is a contraction: ‖〈Q(F)〉T ‖ � ‖Q(F)‖,• norm estimate: ‖Q(F)‖ � Cr‖F‖r , and• 〈Q(F)〉T = Q( 1

T

∫ T

0 Ft dt),

imply that in order to prove the lemma we need to show that the norm limit of1T

∫ T

0 Ft dt in Cr(TN × R

N) is 〈F 〉.Let us write down the formulas for the Fourier components of Ft and 〈F 〉.

Formula (12) gives:

Ft (k, p) = eit (k2/2+kp)F (k, p).

It follows that

1

T

∫ T

0Ft (k, p) dt =

F (k, p), if kp + k2/2 = 0,

eiT (kp+k2/2) − 1

iT (kp + k2/2)F (k, p), otherwise.

Formula (13) implies:

〈F 〉(k, p) ={

F (k, p), if kp + k2/2 = 0,

0, otherwise.

We can now estimate the norm of the difference:∥∥∥∥ 1

T

∫ T

0Ft dt − 〈F 〉

∥∥∥∥r

= sup{k,p∈ZN : kp+k2/2=0}

(1 + k2)r/2

∣∣∣∣(eiT (kp+k2/2) − 1)

iT (kp + k2/2)

∣∣∣∣|F (k, p)|

� 2

T‖F‖r sup

{k,p∈ZN : kp+k2/2=0}|(kp + k2/2)−1| � 4

T‖F‖r ,

where in the last step we used the fact that kp + k2/2 is half-integer. Clearly thisimplies that 1

T

∫ T

0 Ft dt converges in norm to 〈F 〉 and the thesis follows. �Let, as before, µn = n2/2 be the eigenvalues of H . We use the notation N(E) :=

#{µn � E} for the number of eigenvalues of H which are less or equal E. Recall[12] that an operator a is called semi-classically negligible (SN) with respect to H

if

τ(a∗a) := limE→∞ τE(a∗a) := lim

E→∞1

N(E)

∑µn�E

(φn, a∗aφn) = 0.

PROPOSITION 9. Semi-classically negligible operators form a closed left-sidedideal in the algebra of all bounded operators.


Proof. We need to verify the following three things:

1. If a, b are SN so is a + b.2. If ak → a and ak are SN then a is also SN.3. If a is SN and b bounded then ba is SN.

Since (φn, a∗aφn) = ‖aφn‖2, part 1 follows from the estimate:

‖(a + b)φn‖2 � 2(‖a0φn‖2 + ‖bφn‖2). (14)

To prove part 2 we set up a 2ε estimate. Using (14) we get:

1

N(E)

∑µn�E

‖aφn‖2 � 2

N(E)

∑µn�E

‖(a − ak)φn‖2 + 2

N(E)

∑µn�E

‖akφn‖2

� 2‖(a − ak)‖2 + 2

N(E)

∑µn�E

‖akφn‖2,

and both terms are small. Finally, part 3 is obvious:

‖baφn‖2 � ‖b‖2‖aφn‖2. �The following is the main result of our paper:

THEOREM 10. With the above notation

〈Q(F)〉 = Q(F ) + aF ,

where

F (x, p) = F (0, p) = 1

(2π)N

∫TN

F (x, p) dNx,

and aF is a semi-classically negligible operator. Moreover, if N = 1, then aF is acompact operator.

Proof. We can rephrase the theorem as saying that all but zero Fourier compo-nent of F give rise to operators with semi-classically negligible ergodic averages.Also notice that typically if F (0, p) does not go to 0 as p → ∞ then Q(F ) is notSN.

Items 1 and 2 in Proposition 9 let us use an approximation argument. Indeed itis enough to verify that for functions in the following form: F(x, p) = eikxg(p),k �= 0, the corresponding ergodic averages 〈Q(F)〉 = Q(〈F 〉) are SN.

It follows from (13) that for such F we have

〈F 〉(x, p) ={

eikxg(p), if kp + k2/2 = 0,

0, otherwise.

Since Q(〈F 〉)φn(x) = 〈F 〉(x, n) einx we have

Q(〈F 〉)φn(x) ={

ei(k+n)xg(n), if kn + k2/2 = 0,

0, otherwise.(15)


We can now estimate τE(Q(〈F 〉)∗Q(〈F 〉)) as follows:

1

N(E)

∑µn�E

‖Q〈F 〉φn‖2 = 1

N(E)

∑{n∈ZN : µn�E, kn+k2/2=0}

|g(n)|2

� ‖F‖r

#{n ∈ ZN : 1

2n2 � E, kn + k2/2 = 0}

#{n ∈ ZN : 12n

2 � E} .

In the above expression the denominator #{n ∈ ZN : 1

2n2 � E} behaves like EN/2

for large E, while the numerator, because of the constraint kn+ k2/2 = 0 grows atmost like EN/2−1, if N > 1. Consequently,

1

N(E)

∑µn�E

‖Q(〈F 〉)φn‖2 = O(1/E),

which proves that Q(〈F 〉) is SN. Consider now the case of N = 1. We can againuse an approximation argument and, just as before, it is enough to verify that forfunctions in the following form: F(x, p) = eikxg(p), k �= 0, the correspondingergodic averages 〈Q(F)〉 = Q(〈F 〉) are compact. Examining formula (15) we seethat Q(〈F 〉)φn(x) is nonzero only when kn + k2/2 = 0. But this equation has nosolutions if k is odd, and one solution n = −k/2 if k is even. In any case we obtaina finite-dimensional operator. Since norm limits of finite-dimensional operators arecompact, the theorem is proved. �

References

1. Anosov, D. V.: Geodesic flows on closed Riemann manifolds with negative curvature, Proc. ofthe Steklov Institute of Mathematics 90 (1967).

2. Colin de Verdiere, Y.: Ergodicite et fonctions propres du Laplacien, Comm. Math. Phys. 102(1985), 497–502.

3. Jakobson, D.: Quantum limits for flat tori, Ann. Math. 145 (1997), 235–266.4. Klimek, S. and Lesniewski, A.: Quantized Kronecker flows and almost periodic quantum field

theory, J. Math. Phys. 38 (1997), 5605–5625.5. Klimek, S. and Lesniewski, A.: Quantum maps, In: L. Coburn and M. Rieffel (eds), Proceed-

ings of the 1996 Joint Summer Research Conference on Quantization, Contemp. Math. 214,Amer. Math. Soc., Providence, RI (1997).

6. Klimek, S. and Lesniewski, A.: Quantum ergodic theorems, In: L. Coburn and M. Rieffel,Proceedings of the 1996 Joint Summer Research Conference on Quantization, Contemp. Math.214, Amer. Math. Soc., Providence, RI (1997).

7. Klimek, S., Lesniewski, A., Maitra, N. and Rubin, R.: Ergodic properties of quantized toralautomorphisms, J. Math. Phys 38 (1997), 67–83.

8. Rudnick, Z. and Sarnak, P.: The behavior of eigenstates of arithmetic hyperbolic manifolds,Comm. Math. Phys. 161 (1994), 195–213.

9. Schnirelman, A.: Ergodic properties of the eigenfunctions, Usp. Math. Nauk 29 (1974), 181–182.

10. Zelditch, S.: Quantum ergodicity of C∗ dynamical systems, Comm. Math. Phys. 177 (1996),

507–528.


11. Zelditch, S.: Quantum transition amplitudes for ergodic and for completely integrable systems,J. Funct. Anal. 94 (1990), 415–436.

12. Zelditch, S.: Quantum dynamics from the semi-classical point of view – unpublished notesavailable on the Web.

13. Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces, DukeMath. J. 55 (1987), 919–941.

14. Zelditch, S.: The averaging method and ergodic theory for pseudo-differential operators oncompact hyperbolic surfaces, J. Funct. Anal. 82 (1989), 38–68.


Huygens’ Principle, Dirac Operators, and RationalSolutions of the AKNS Hierarchy

FABIO A. C. C. CHALUB and JORGE P. ZUBELLIIMPA, Est. D. Castorina 110, RJ 22460-320, Brazil. e-mail: {fabio, zubelli}@impa.br

(Received: 16 September 2003; in final form: 6 May 2004)

Abstract. We prove that rational solutions of the AKNS hierarchy of the form q = σ/τ and r = ρ/τ ,where (σ, τ, ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens’ type,i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthensthe connection between the theory of completely integrable systems and Huygens’ principle byextending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical resultof Lagnese and Stellmacher concerning perturbations of wave operators.

Mathematics Subject Classifications (2000): 37K10, 35Qxx, 35B40.

Key words: Dirac operators, fundamental solutions, Huygens’ principle, integrable hierarchies,rational solutions.

1. Introduction

Music and other familiar features of signal propagation, as we know it in our expe-rience, would be very different if we lived in an even-dimensional space. Indeed,one of the striking characteristics of the fundamental solutions to the wave equation

�ψdef=

(∂2

0 −n∑

i=1

∂2i

)ψ = δy, (1)

in n = 3 space dimensions is the fact that the support of the fundamental solutionψ(·; y) is the surface of the light-cone with vertex at y, and not the union of thissurface with the interior part thereof. This implies that a sharp signal of finiteduration will not be ‘heard or felt’ by an observer after a sufficiently long elapsedtime. This amazing property is shared by the wave equation for n = 3, 5, 7, . . . butis not shared for n even or n = 1.

J. Hadamard, in his celebrated Yale Lectures [23], called the above propertyHuygens’ minor premise. He further studied it for perturbations of the wave opera-tor of the form �+u, where u = u(x0, x1, . . . , xn). He characterized such propertyin terms of the vanishing of the logarithmic term of his elementary solution. How-ever, such simple-looking conditions are very elusive since they hide complicatednonlinear partial differential equations for the perturbation u. In fact, such con-ditions were so strong that for a while it was conjectured that for n = 3, 5, . . .

188 FABIO A. C. C. CHALUB AND JORGE P. ZUBELLI

and modulo certain trivial transformations only the unperturbed D’Alembertianpossessed Huygens’ property. In this context, a trivial transformation means thecomposition of changes of the independent variable, and left or right multiplicationof the operator by nonzero functions. For an early account, the reader is referred toCourant and Hilbert’s masterpiece [13].

It came as a striking surprise the discovery by Stellmacher [40, 41] of Huygens’type wave operators of the form �+u that were not trivially equivalent to �. How-ever, as most of the fundamental discoveries in mathematical physics, such findingwas only the tip of an iceberg. It turned out that the potentials found by Stellmacherwere actually the first ones of a full family of potentials later explored by Lagneseand Stellmacher [31] that yield Huygens’ type operators. The construction of sucha family was accomplished through a technique, namely Darboux transformation,which was independently used in soliton theory to construct the rational solutionsof the Korteweg–de Vries (KdV) equation:

V = Vxxx − 6V Vx.

The fact that the KdV equation possesses large families of solutions that remainrational in the variable x by its flow is already a deep one and is directly linked withseveral other miracles that such equation displays. For an early account see [2, 3],whereas a modern approach can be found in [45]. In fact, it turned out that the find-ings of Lagnese and Stellmacher remained unknown to the integrable system com-munity for a while and were only pointed out much later by R. Schimming [38].

Given the beauty and the importance of Huygens’ property for the meaningfultransmission of information, it is natural to ask the question: which other hy-perbolic equations of physical relevance share such a property? This has beenthe subject of attention by several authors, see, for example, [5, 22, 26, 27] andreferences therein.

The purpose of the present work is to consider Huygens’ property for perturbedDirac operators in Minkowski space. Such operators act on (generalized) functionswhose image belongs to a suitable Euclidean space. Unperturbed Dirac opera-tors for curved pseudo-Riemannian metrics have been studied, using a differentapproach, by P. Günther, V. Wünsch, and their collaborators. See the referencesin [22] to those authors.

Dirac operators on Lorentzian spin manifolds were studied by H. Baum, wheresome examples of Dirac operators of Huygens’ type were presented. See [4].

More precisely, we consider the fundamental solutions � of Dirac operatorsD� = δy , where as defined in Section 3,

D =∑

µ

γ µ∂µ + v. (2)

We say that D satisfies Huygens’ property (in Hadamard’s strict sense) or is ofHuygens’ type if for every y the domain of dependence of the distribution �(·; y)

is precisely the light-cone with vertex in y.

HUYGENS’ PRINCIPLE AND DIRAC OPERATORS 189

Our main result is that by taking for (q, r) certain scalar fields that are rationalsolutions of the AKNS hierarchy of nonlinear equations and setting

D = �∂ − 12(q + r)I + 1

2(q − r)γ ,

where I is the identity, γ is a certain matrix and �∂ is the free-space Dirac operatorin an even-dimensional spacetime of sufficiently high dimension, the operator D isof Huygens’ type. Such fields q and r have explicit expressions in terms of Schurpolynomials.

The AKNS hierarchy of nonlinear evolution equations is an infinite family ofcommuting Hamiltonian flows generated by zero curvature conditions of certainmatrix systems according to the Lax pair formalism. Furthermore, this hierarchy isa completely integrable system and generalizes several important integrable equa-tions including the KdV hierarchy [1]. It includes as particular cases the cubicnonlinear Schrödinger equation, the modified KdV equation, and others.

Our result extends to the context of Dirac operators and the rational solutionsof the AKNS hierarchy the result of Lagnese–Stellmacher for the wave operatorand the rational solutions of the KdV hierarchy. This extension turns out to be in ahighly nontrivial direction since the operators D so obtained when squared do notyield (in general) diagonal operators. However, for very particular choices of theparameters that enter the fields q and r we obtain as consequence of our result thealready known result of Lagnese and Stellmacher.

The chain of unexpected connections does not stop in the link between the ratio-nal solutions of the AKNS hierarchy and Dirac operators of Huygens’ type. Theyalso manifest themselves in the so-called bispectral property which was introducedby Grünbaum, and analyzed by Duistermaat and Grünbaum in [15]. It turns out thatthe fields (q, r) that yield Dirac operators of Huygens’ type also yield bispectralAKNS operators.

The plan for this article goes as follows:We conclude this introduction with an account on fundamental solutions of

wave operators and Hadamard’s series. This will set up the notation and basicresults that will be needed further in the paper.

Section 2 collects some facts from infinite-dimensional integrable systems thatare needed in the proof of our results. It deals with integrable systems and inparticular with the AKNS hierarchy, which is an important generalization of theKdV hierarchy.

Section 3 is concerned with Dirac operators. Section 4 deals with the analyticproperties of such operators that are needed in Section 5, which contains the proofof the main result.

1.1. HUYGENS’ PRINCIPLE FOR WAVE OPERATORS

Let us start by considering a fundamental solution to the wave equation

�ψ + uψ = δy. (3)


Here, as before, � = ∂20 − ∑n

i=1 ∂2i and u = u(x0, . . . , xn) is a given potential in

n spatial dimensions and time x0.Let λ denote the ‘time-like geodesic distance’

λ ={ √

(x0)2 − ∑ni=1(x

i)2, if (x0)2 − ∑ni=1(x

i)2 � 0,

0 otherwise.

For values of α with positive real part we define the Riesz kernel� as the distribution

�α = N(α)λα, (4)

where the numerical normalization factor is given by

N(α) = 1

2

[2α+nπ(n−1)/2�

(α + n + 1

2

)�( 1

2α + 1)

]−1

. (5)

The numerical factor N(α) obeys the important recursion rule

(α + 2)(α + n + 1)N(α + 2) = N(α). (6)

If the real part of α is sufficiently large, with the help of the preceding formula, oneproves

��α+2 = �α.

From Equation (5) we see that N(α) = 0 when α = −2, −4, −6, . . . or α =−n−1, −n−3, −n−5, . . . . In these cases �α is supported on the light-cone withvertex in the origin. By an analytic continuation of �α to the region �(α) > −n−2,one finds that �−n−1 = δ0, so �−n+1 is the fundamental solution of �. For anarbitrary y, we use the translation invariance of the D’Alembertian.

From the aforementioned expressions for N(α) we conclude that the funda-mental solution of � is supported on the light-cone (and, consequently, � has theHuygens property) if, and only if, n is odd and greater than 1. For the details of theabove calculation, the reader is referred to [18].

In order to construct the fundamental solution � of the operator �+u, we shalldevelop a series expansion in the above defined Riesz kernels,

� =∞∑

k=0

�−n+1+2kwk. (7)

Applying the operator � + u to � and equating to �−n−1 = δy we find thefollowing recursion:

w0 = 1,

wk + 1

k

n∑µ=0

(xµ − yµ)∂µwk = −(� + u)wk−1. (8)

� Note that α is an exponent in λ and an index in �.


This recursion is known as Hadamard’s recursion. Its termination at k, i.e.,wk′ = 0, for every k′ > k, implies validity of Huygens’ principle for n = 2k + 3,as can be readily seen from the Riesz kernels properties stated before.

The termination requirement implies a highly nonlinear condition over u, whichmakes it very difficult to find examples of Huygens’ potentials.

An entire family of Huygens potentials of the form � − uk(x0), k � 1, was

found by Lagnese and Stellmacher [31]. They are trivially equivalent to operatorswith potentials given by

uk(t) = −2d2

dt2log ϑk(t),

where ϑk is known as the kth Adler–Moser polynomial.� Those operators possessHuygens’ property if, and only if, n is odd and n � 2k+3. As remarked in [38, 39]those potentials are the same ones found by Adler and Moser [2] in the frameworkof the KdV hierarchy rational solutions. By rational solutions of the KdV hierarchywe mean those rational functions (in x) that remain rational by the flow of the KdVhierarchy equations.

2. Preliminaries on Integrable Systems

In this section we shall briefly review some of the relevant material on infinite-dimensional integrable systems that will be used in the text. More specifically weshall concentrate on the so-called Ablowitz–Kaup–Newell–Segur (AKNS) hierar-chy of nonlinear evolution equations. This hierarchy consists of an infinite familyof commuting Hamiltonian vector fields on a pair of scalar fields (q, r). We shallnow recall some key facts about the AKNS hierarchy that will be needed in thesequel following closely the construction in [34].

2.1. THE AKNS HIERARCHY

The theory of solitons is extremely rich and connects to a vast number of seeminglyunrelated fields. One of its first developments was the remark that the Korteweg–de Vries equation could be interpreted as a compatibility condition for certainlinear differential equations, one of which was the Schrödinger equation. For ahistorical overview the reader should consult [10, 33, 34] and references therein.

A crucial step in the development of soliton theory resulted from the remarkthat, besides the KdV, a number of other important equations in mathematicalphysics can be expressed as compatibility conditions of linear systems of theform [1, 46]

∂x� = (kH + Q)�def= (kH + qE + rF )�, (9)

� The first reference we know of such polynomials seems to be in the work of J. Burchnall andT. Chaudy [9].


where q and r are certain functions, and

∂t�� = R(�)�def= (k�H + R1k

�−1 + · · · + R�)�. (10)

In the last equation H , E, and F are given by

H = diag[1, −1]and

E = F� =[

0 10 0

].

The matrices R1, R2, . . . are required to be traceless. As we shall see, their expres-sions (in terms of q, r , and their derivatives) are determined from the compatibilityof Equations (9) and (10). For that, we write the coefficient Rl of Equation (10) as

Rldef= elE + flF + hlH (11)

and take e0 = f0 = h1 = 0, h0 = 1, e1 = q, and f1 = r .If we take � = 1, we get ∂t1q = ∂xq and ∂t1r = ∂xr , which are trivially solved in

terms of translations of q and r . This explains the usual convention of identifyingthe variable t1 with x.

The first interesting example occurs when we take � = 2, which is connectedto the cubic Schrödinger equation. The other equations of the AKNS hierarchy canbe found by considering the compatibility conditions of (9) and (10)

∂t�Q − ∂xR(�) + [kH + Q, R(�)] = 0, (12)

viewed as an identity in k. For � = 3, we get

qt3 = 14(qxxx − 6qrqx),

rt3 = 14(rxxx − 6qrrx).

It is immediate to check that in the � = 3 case above, q = ±r gives the modifiedKdV equation. Furthermore, such reduction is compatible with all the odd flows ofthe AKNS hierarchy.

Equation (12) is equivalent to

el+1 = qhl + 12∂xel, (13)

fl+1 = rhl − 12∂xfl, (14)

∂xhl+1 = fl+1q − el+1r, (15)

with 0 � l � n − 1, together with

∂tl q = ∂xel + 2qhl, (16)

∂tl r = ∂xfl − 2rhl. (17)


The coefficients ej , hj and fj can now be viewed as polynomials in the variables∂lxq and ∂l

xr , with l ∈ Z�0.As a consequence of the choice of

hl+1 = −1

2

∑m + n = l + 1

m, n � 1

(enfm + hnhm), (18)

justified in [50], it follows that the AKNS equations can be written down using therecursion (13), (14) and (18). Furthermore, if we define el , fl , hl for all values ofl ∈ Z>0, the �th equation in the AKNS hierarchy can be written as

qt� = 2e�+1(q, . . . , ∂�xq; r, . . . , ∂�−1

x r),

rt� = −2f�+1(q, . . . , ∂�−1x q; r, . . . , ∂�

xr).(19)

2.2. HIROTA VARIABLES AND SCHLESINGER TRANSFORMATIONS

One way of constructing large classes of solutions to the AKNS hierarchy is bymeans of Schlesinger transformations [17, 28, 29, 34]. In very broad terms, the keyidea is to introduce a transformation on the general solution � of Equations (9)and (10) of the form � = S�. Such transformation induces a change on thematrices Q and R(�) that preserves the form of Equation (12), namely

R(�) = SR(�)S−1 + St�S−1. (20)

Furthermore, if det(S) is independent of the AKNS time variables, then the tracesof the matrices Q and R(�) are preserved. Hence, if we start with Q and R(�) of theform given in Equation (11) the transformed variables are also of such form. Thisin turn implies that the transformed fields, which we will denote by (q, r), satisfyequations of the form (19). In particular, we have

R(1) = kH + qE + rF.

In the sequel, we shall make use of the Schlesinger transformations induced bygauge transformations of the form

S =[ −2k − ∂t1 log(e1) e1

1/e1 0

]. (21)

This induces at the level of Hirota variables the transformation�

S: (σ, τ, ρ) �−→ (σ+, τ+, ρ+)def=

(τ, −ρ, −ρxxρ − ρ2

x

τ

)(22)

� We remark for the benefit of readers of [34] that we have deviated a bit from its notation here.To transform from its variables to ours we have to perform the following changes: ρ �→ σ , σ �→ −ρ,τ �→ τ , q �→ r , r �→ −q, x �→ ix, where the variables on the left are the ones used in [34].


and at the level of fields e1 and f1

S: (e1, f1) �−→ (e1, f1) =(

− 1

f1, f1 xx − f 2

1 x

f1− f 2

1 e1

). (23)

Schlesinger transformations will be used in the next section to generate familiesof rational solutions to the AKNS hierarchy, and will also be instrumental in theproof that the corresponding Dirac operators are of Huygens’ type.

2.3. SCHUR POLYNOMIALS AND RATIONAL SOLUTIONS OF AKNS

Our construction of Huygens’ type Dirac operators starts at the level of Hirota vari-ables [34]. For that, we introduce the infinite sequence of the so-called elementarySchur polynomials

{Qj (y1, y2, . . . , yj )}∞j=0

defined by the requirement that∑j�0

λjQj = exp

(∑j�1

λjyj

). (24)

For convenience, we extend the definition of Schur polynomials as zero for nega-tive indexes. Using this convention it is easy to check that

∂ykQj = Qj−k. (25)

We introduce the Wronskians

τ dj = W[Qd, Qd−1, . . . , Qd−j ], (26)

where the derivatives should be interpreted with respect to the first variable y1. Weset y1 = x and take

q = τ dj−1

τ dj

, r = −τ dj+1

τ dj

. (27)

For convenience, we extend the definition (26) of τ dj as follows:

τ dj

def={

1 if j = −1 and d = 0, 1, . . . ,0 if j < −1 or d < 0.

We shall now describe how Equation (26) yields a set of rational solutions of theAKNS hierarchy.

In order to to do that, we relate the variables yk in the Schur polynomials withthe time variables tk of the AKNS hierarchy by setting

yk = 1

(−2)k−1tk, k = 1, 2, . . . . (28)

We now quote the following:


THEOREM 1 (R. Sachs [35]). For every pair of integers d and j , such that d �j > −1, the pair of fields (q, r) defined by Equations (26), (27), and (28) is arational solution of the AKNS hierarchy.

An alternative proof of Theorem 1 can be found in Section 5 of [50].It turns out that the AKNS solutions defined by (27) and (28) are associated to

the Hirota variables (σ, τ, ρ) = (τ dj−1, τ

dj , −τ d

j+1). See [34].We can interpret, still following [34], the Schlesinger operator as a shift operator

in j in the sequence {τ dj }∞j=−∞ (as can be seen in Equation (22)), S[τ d

j ] = τ dj+1. If

we consider j = −1, then (q, r) = (0, −Qd), which is a solution of the AKNS hi-erarchy. The effect of each successive Schlesinger transformation on the pair (q, r),produces another rational solution for the AKNS hierarchy of the form in Equa-tion (27). The new solution (q, r) is related to the previous one by Equation (23),i.e.,

(q, r) =(

−1

r, rxx − r2

x

r− r2q

).

We close with a remark that will be used to prove some of the results of Lagneseand Stellmacher as a consequence of ours.

The odd flows of the AKNS hierarchy reduce to the mKdV flows if we imposeq = r . Furthermore, the celebrated Miura transformation u = qx ± q2 sendssolutions of the mKdV hierarchy into solutions of the KdV hierarchy. This circleof ideas is completed by the connection between some of the rational solutions ofthe AKNS we just constructed and the Adler–Moser polynomials.

Remark 1 ([12]). If y2k = 0 for all k ∈ N, then

τ2j+1j+1

τ2j+1j

(x, 0, y3, 0, y5, . . .) = (−1)j+1∂x logϑj+1

ϑj

(x, χ2, χ3, . . .),

where ϑj is the j th Adler–Moser polynomial and for a suitable choice of the freevariables y3, y5, . . . and χ2, χ3, . . . .

3. Preliminaries on Dirac Operators

The goal of the present section is to review the definition of Dirac operators anddiscuss some of their properties. In order to do that we start with a brief descriptionof Clifford algebras.


3.1. CLIFFORD ALGEBRAS

Let {γ µ}, µ = 0, . . . , n, be a set of elements in an associative algebra over the fieldof real numbers such that

{γ µ, γ ν} def= γ µγ ν + γ νγ µ = 2gµνI, (29)

where (gµν) = diag[1, −1, −1, . . . , −1] is the Minkowski tensor in n + 1 dimen-sions and I is the algebra’s identity. It shall be henceforth omitted.

The Clifford algebra generated by these elements is the set of all linear combi-nations of all products in the form

(γ 0)m0(γ 1)m1 · · · (γ n)mn, (30)

where mµ = 0 or mµ = 1 and when all mµ = 0 the above product should beinterpreted as the identity. If n is odd the Clifford algebra has maximum dimension2n+1. So, all the products in the form (30) are linearly independent. See [19, 32].

If n is odd we can define a matrix γ such that {γ , γ µ} = 0, µ = 0, . . . , n andγ 2 = 1. We remark that such a matrix cannot be defined for n even. The aboverelations uniquely define, modulo a sign, γ = (−1)(n−1)/4γ 0γ 1 · · · γ n.

A particular representation of this algebra can be obtained in terms of 2N × 2N

complex matrices, where N = (n + 1)/2. The matrices representing the γ µ arecalled Dirac matrices. Due to Pauli’s fundamental theorem [36, 43] all the possiblerepresentations are conjugated.

We define the Weyl representation, where

γ 0 =(

0 I

I 0

), γ i =

(0 −σ i

σ i 0

), γ =

(I 00 −I

).

In dimension 3 + 1, the σ i are the celebrated Pauli matrices.From now on, summation for repeated indices is implied, from 0 to n for Greek

indices and from 1 to n for Latin indices.

3.2. DIRAC OPERATORS

Following the notations and definitions introduced in Subsection 3.1, Dirac opera-tors are defined by [42]

D = γ µ∂µ + v.

We shall also adopt the notation �∂ = γ µ∂µ and restrict ourselves to the case wherev is a linear combination of I and γ ,

v = aI + aγ .

We call a the scalar potential and a the pseudo-scalar potential.


The attentive reader may ask about the complex unit i that appears on the sideof the free Dirac operator �∂ in many textbooks. In Theorem 5 we will see that thisis immaterial for our purposes.

It is easy to see that

�∂2 = �. (31)

4. Dirac Operators and Huygens’ Principle

4.1. HUYGENS’ PRINCIPLE REVISITED

By a fundamental solution to the Dirac operator we mean a solution � of

(�∂ + v)� = δy,

where δy denotes Dirac-delta distribution supported in an arbitrary point y in space-time.

We shall say that a Dirac operator �∂ + v (and by extension, its fundamentalsolution) obeys Huygens’ principle if � satisfies

supp � ⊂ C(y),

for all y ∈ Rn, where

C(y) = {x | (xµ − yµ)(xµ − yµ) = 0},is the light-cone with vertex in y.

More generally, for a linear hyperbolic differential operator L with matrix coef-ficients, we shall say that it obeys Huygens’ principle in a causal neighborhood �

if the support of its fundamental solution is contained in the light-conoid associatedto the highest-order coefficients of L.

4.2. FREE DIRAC OPERATORS

From Equation (31) we can see that �∂�α is a fundamental solution of the free Diracoperator �∂ whenever �α is a fundamental solution of the wave operator.

This allows the definition of a family of distributions, defined in a similar wayas the Riesz kernels �α were defined in Subsection 1.1, namely

�α def= �∂�α. (32)

We shall call �α a Dirac kernel. We easily verify that �∂�α = �α−2.Let us define

�def= γ µ(xµ − yµ).


We have that, for λ(x − y) > 0

�2 = λ2. (33)

We are now ready to determine which free Dirac operators satisfy Huygens’property.

THEOREM 2. The free Dirac operator in dimension n obeys the Huygens’ prin-ciple if, and only if, n is odd.

Proof. We recall that for the real part of α large enough

�α = �∂�α = αN(α)λα−2�.

From here, by means of straightforward manipulations on Riesz kernels, we get that�α is supported in the light-cone if, and only if, α is a zero of the normalizationfactor αN(α), i.e, if, and only if, α = 0, −2, −4, −6, . . . or α = −n − 1, −n − 3,

−n− 5, . . . . As �−n+1 is the fundamental solution of �∂ in a space of dimension n,we conclude that �∂ possesses the Huygens property if, and only if, n is odd. �

Further discussions on free Dirac operators appear in [11].

4.3. HADAMARD EXPANSIONS

Let � denote the solution of

(�∂ + v)� = δy. (34)

Adapting Hadamard’s seminal idea for the wave operator, we look for a seriesexpansion of � of the form

� =∞∑

k=0

{�α0+2ks2k + �α0+2ks2k+1}, (35)

where sk = sk(x, y) is a matrix coefficient and α0 should be taken equal to −n+1.As v and �α are matrices, we need to know how to exchange these two objects.We conclude that

v�α = �αv∗, (36)

where

v∗ def= aI − aγ . (37)

Equating (�∂ + v)� = �−n−1 we find a recursion akin to Hadamard’s:

s0 = 1, (38)

s2k+1 = (�∂ − v∗)s2k, (39)

s2k + 1

k(xµ − yµ)∂µs2k = −(�∂ + v)s2k−1. (40)


THEOREM 3. If v = aI + aγ , � is given by Equation (35), and satisfies(�∂ +v)� = δy , then the coefficients in the expansion (35) are uniquely determinedby imposing the regularity of sk in the vicinity of the vertex of the light-cone, i.e,when x → y.

Proof. The uniqueness for odd terms is obvious. For even terms, suppose thats1 and s2 are solutions of the third equation in the system (38), (39) and (40). So,their difference s = s1 − s2 is a solution of

s + 1

k(xµ − yµ)∂µs = 0.

Consider a ray emanating from the light-cone vertex y and remember that λ mea-sures the geodesic distance between x and y. Studying the behavior of s along thisray we find that s obeys the equation

s(λ) + 1

kλs ′(λ) = 0.

The only nontrivial solution of this equation is given by s ∝ λ−k, which is notwell-behaved near λ = 0. Thus, s = 0 and the theorem is proved. �THEOREM 4. Let n � 1 be an odd integer and n � d ∈ N. If the Hadamardcoefficients sk for the operator �∂ +v vanish for all k � d, then �∂ +v is a Huygens’type operator.

Proof. Just note that all the Riesz and Dirac kernels that appear in the solutionexpansion are of Huygens’ type. �

Our next step is to consider transformations that preserve Huygens’ principlefor Dirac operators in trivial way. A few examples are:

(i) Change of independent variables by a smooth diffeomorphism: xµ = f µ(x0,

. . . , xn), µ = 0, . . . , n, with det(∂µf ν)µ,ν=0,...,n �= 0.

(ii) Left multiplication: take D �→ D = �(x)D and � �→ � = ��(y)−1,where � ∈ C1(Rn+1; R

N×N) and �(x) is a nonsingular matrix for all x.(iii) Factor transformations: let ρ be a nonsingular smooth matrix-valued function,

i.e.,

ρ = ρφI + ρµγ µ + ρµγ µγ + ργ ,

where ρφ , ρµ, ρµ and ρ are smooth functions with det(ρ(x)) �= 0, for all x

in the domain under consideration. The factor transformation consists in themapping D �→ D = ρ(x)Dρ(x)−1 and � �→ � = ρ(x)�ρ(y)−1.

THEOREM 5. The transformations (i)–(iii) above preserve the validity of Huy-gens’ principle.


Proof. In the case of a diffeomorphic changes of variable, the claim followsfrom the fact that the light-conoid will change accordingly. In both remaining casesthe fundamental solutions �(x; y) of the transformed operators can be written interms of the fundamental solutions �(x; y) of the original operators. The entriesof �(x; y) being linear combinations of the entries of �(x; y) with coefficients inthe field of smooth functions of x. Thus, for each y if the support of �(x; y) iscontained in C(y), so is the support of �(x; y). �

Two operators related by the transformations (i)–(iii) above are called equiva-lent (in the sense of Theorem 5).

PROPOSITION 1. Changes of representation of Dirac matrices do not affect Huy-gens’ property.

Proof. Due to Pauli’s Fundamental Theorem [36, 43] all the representations areequivalent. �

Remark 2. For any potential of the form v = a(x0)I + a(x0)γ , the operators�∂ + v, �∂ − v, �∂ − v∗ and �∂ + v∗ are equivalent in the sense of Theorem 5.

5. Main Result

In this section we shall prove the main result of the present work. In order to dothat we start with a preliminary lemma:

LEMMA 1. Let v = v∗ = 0 and g ∈ C∞(Rn+1). Suppose {sk}∞k=1 is the sequenceof regular (near the light-cone) solutions of the the recursion defined by Equa-tions (39), (40), and starting from s1 = −g. Then, the odd terms are given, fork = 1, 2, . . . , by

s2k+1(x, y) = (−1)k+1k!∫ 1

0· · ·

∫ 1

0�k g(ξk)z

k1z

k+12 · · · z2k−1

k dz1 · · · dzk, (41)

where the sequence of variables {ξi}∞i=0 is defined recursively by ξ0 = x and ξi =y + zi(ξi−1 − y) for i = 1, 2, . . . .

Proof. We notice that

(xµ − yµ)∂

∂xµF (y + z(x − y)) = z

dF

dz(y + z(x − y)).

This and the regularity near the light-cone allows us to write the solution s2k ofEquation (40) with v = 0 as

s2k(x, y) = −k

∫ 1

0�∂s2k−1(y + z(x − y), y)zk−1 dz, k � 1.


From Equation (39) it follows that

s2k+1 = �∂s2k = −k

∫ 1

0� s2k−1(y + z(x − y), y)zk dz, k � 1,

and a simple induction gives (41). �LEMMA 2. If f (x) satisfies �p f (x) = 0 for some positive integer p, then theseries for �∂ + f (x)(I + γ ) truncates.

Proof. The central idea here is the fact that we can construct a new recursionfrom the Hadamard recursion associated to �∂ +f (x)(I + γ ) such that the functionf appears only on the initial condition of such new recursion and then apply theprevious lemma.

Consider two sequences, {sk}∞k=0 and {sk}∞k=1, solving Equations (39) and (40)associated to v = f (x)(I + γ ) and to v = 0, respectively, and starting from s0 = 1and from s1 = −f , respectively.

Firstly, we prove that sk = (I + (−1)kγ )sk, for k � 1. This will be shown byinduction. The case k = 1 is trivial. To show that the validity of the claim for 2k−1implies it for 2k, we consider Equation (40). From the anti-commutation relations,we get

s2k + 1

k(xµ − yµ)∂µs2k − (I + γ )

(s2k + 1

k(xµ − yµ)∂µs2k

)= −(�∂ + f (I + γ ))s2k−1 + (I + γ )�∂s2k−1 = 0.

The regularity in the vicinity of the light-cone’s vertex yields the claim.To show that the validity of the claim for 2k implies it for 2k + 1, we use

Equation (39) and the anti-commutation relations. Indeed, we write

s2k+1 − (I − γ )s2k+1 = (�∂ − f (I − γ ))s2k − (I − γ )�∂s2k = 0.

To conclude the proof we notice, using Lemma 1, that

s2k+1(x, y) = (−1)k+1k!∫ 1

0· · ·

∫ 1

0�k f (ξk)z

k1z

k+12 · · · z2k−1

k dz1 · · · dzk.

Thus, the Hadamard coefficients

s2k+1 = (I − γ )s2k+1,

s2k+2 = −(I + γ )(k + 1)

∫ 1

0�∂s2k+1(y + z(x − y), y)zk−1 dz

of �∂ + f (x)(I + γ ) vanish for k � p. �Remark 3. For the particular choice of the potential v = f (I + γ ) we found

that surprisingly the coefficients depend linearly on f . This fact comes from the


identity (I + γ )(I − γ ) = (I − γ )(I + γ ) = 0, which implies that v∗v = vv∗ = 0.So, we re-write Equations (39) and (40) as

s2k + 1

k(xµ − yµ)∂µs2k = −(� − �∂v∗)s2k−2,

with s0 = 1. Then, s2k is a linear function of (scalar) derivatives and powers of�∂v∗ (and products of both). But �p �∂v∗(x)�q �∂v∗(y) = 0, for every x, y ∈ R

n+1,p, q � 0, as a consequence of the property of the gamma matrices. So, s2k is actu-ally linear in �∂v∗. From the Equation (39) and from the fact that v∗(x)�p �∂v∗(y) =0, for every x, y ∈ R

n+1, p � 0, we also conclude that the odd terms are linear in�∂v∗.

COROLLARY 1. If q(x) is a degree-d polynomial, then the Hadamard’s series of�∂ + q(x0)(I + γ ) is truncated at d + 1, i.e., sd ′ = 0, whenever d ′ � d + 2.

Proof. We rewrite Equation (41) as

s2k+1(x, y) = (−1)k+1k!∫ 1

0· · ·

∫ 1

0q(2k)(ξk)z

k1z

k+12 · · · z2k−1

k dz1 · · · dzk,

with ξi = y0 + zi(ξi−1 − y0), and ξ0 = x0. If d is odd, then sd+2 = 0. If d iseven, sd+1 does not depend on x0, sd+2 + 2

d+2(x0 − y0)sd+2 = −�∂sd+1 = 0, then

sd+2 = 0. �Remark 4. Corollary 1 remains valid for the null polynomial if one takes its

degree to be −1.

Remark 5. In order to illustrate the above lemmas, we explicitly compute theHadamard’s sequence associated to the potential v = −((x0)2 + c)(I + γ ), wherec is a constant. Then, s0 = 1, s1 = −(I − γ )((x0)2 + c)), s2 = −2(I + γ )(γ 0x0),s3 = (I − γ ) and sk = 0, k � 4.

LEMMA 3. Schlesinger transformations do not change the number of terms in theHadamard’s series.

Proof. We shall adopt the Weyl representation. In this case the matrices definedin Subsection 2.1 are

H = γ , E = γ 0 I − γ

2, F = γ 0 I + γ

2.

From the construction of the Schlesinger transformation in Subsection 2.2, we seethat R(1) = kH + e1E + f1F implies R(1) = kH + e1E + f1F , where as inEquation (23),

e1 = − 1

f1, f1 = f1 t t − f 2

1 t

f1− f 2

1 e1. (42)


Here, all the functions depend on t = x0. Let us adopt the following notation

R(�)(k) = SS[R(�)(k)],where

SS[A] = StS−1 + SAS−1.

It is easy to see from Equation (42) and from the definition of R(1) and R(1) that

SS[−R(1)(0)] = −R(1)(0) = −SS[R(1)(0)]. (43)

In the sequel, we fix and omit k = 0. Let us take t1 = x0 =def t and assume that,for � �= 1, t� is constant.

We define R and R′ by requiring that

Sγ (γ ∂t − R(1)) = (γ ∂t − R)γ S,

γ S(γ ∂t − R(1)) = (γ ∂t − R′)Sγ ,

where S is given by (21). From the first one, we get

R = SS[γ R(1)]γ ,

and from the second one

γ R′ = SS[R(1)γ ].Using that Eγ = −γ E and the same for F , we find γ R(1) = −R(1)γ and the samefor R′. With the help of Equation (43), we find that R′ = R. Thus, subtracting bothequations

(Sγ − γ S)(γ ∂t − R(1)) = (−γ ∂t + R)(Sγ − γ S).

As S is a linear combination of I , H , E and F , or in other words of I , γ , γ 0, andγ 0γ , we find that

(Sγ − γ S)γ γ 0γ i = γ γ 0γ i(Sγ − γ S).

From this, we conclude that

(Sγ − γ S)(γ ∂t − R(1) + γ γ 0γ i∂i)

= (−γ ∂t + R + γ γ 0γ i∂i)(Sγ − γ S), (44)

where

R = SS[γ R(1)]γ .

Note that (Sγ − γ S) is nonsingular. The operator (γ ∂t − R(1) + γ γ 0γ i∂i), withR(1) = −γ γ 0v, is equivalent in the sense of Theorem 5 to �∂ + v. In the same way,−γ ∂t + R + γ γ 0γ i∂i is equivalent to �∂ + v with v = γ γ 0R. Indeed, we recall that


the potentials depend only on t , invert the space coordinates of −γ ∂t+R+γ γ 0γ i∂i ,and then multiply �∂ + v by −γ γ 0. For v = aI + aγ , we have

γ 0v = (a − a)E + (a + a)F

and similarly for v.Now, we define

e1 = a − a, (45)

f1 = a + a, (46)

and we find after a Schlesinger transformation that

a = − f1 + e1

2, (47)

¯a = − f1 − e1

2. (48)

From Equation (44), it follows that the Schlesinger transformation is a Huygenspreserving transformation (see Theorem 5). �

Remark 6. The previous Lemma gives us an algorithmic way to produce newHuygens’ potentials from given ones.

THEOREM 6. If q(x0) and r(x0) are the solutions of the AKNS hierarchy givenby (27), then

�∂ − q + r

2I + q − r

2γ (49)

has the Huygens’ property in dimension d + 2, if d is odd, or d + 3 if d is even.Proof. Let us consider, in the Hirota variables, j = −1. Then, according to (27),

q = 0 and r = −Qd , the dth elementary Schur polynomial. In other words,

v = Qd(x0)

2(I + γ ).

From Corollary 1 we conclude that Hadamard’s series of �∂ +v terminates at d +1.From Lemma 3, we know that a Schlesinger transformation does not change theorder of the series. We also know that a Schlesinger transformation adds one tothe Hirota’s variable j index [34]. Successive transformations go through j =0, . . . , d. Then we prove that the Hadamard’s series of the Dirac operator givenby (49) terminates at d + 1, i.e., sd ′ = 0, d ′ � d + 2. From Theorem 4 it followsthat the operator in Equation (49) is Huygens in dimension d + 2 if d is odd andd + 3 if d is even. �


5.1. ILLUSTRATIVE EXAMPLES

In this subsection, we shall adopt x0 = t and y0 = t0. The constants tk, k = 2, 3, . . .

are arbitrary.

EXAMPLE 1. If d = 3 and j = 1, we have the operator

�∂ + 3t3 + 2t3

t4 − 3t3t + 3t22

I + 6t2t

t4 − 3t3t + 3t22

γ .

We immediately see that if we impose t2 = 0, we have a scalar potential.

EXAMPLE 2. An interesting example is given if we choose d = 2 and j = 2:

�∂ + 1

2

2 + t2 − t2

t2 + t2I − 1

2

t2 − t2 − 2

t2 + t2γ .

Its fundamental solution is given by

� = �−n+1 −− �−n+1

[1

2

2 + t2 − t2

t2 + t2I + 1

2

t2 − t2 − 2

t2 + t2γ

]+

+ �−n+3

[2

t0t − t2

(t2 + t2)(t20 + t2)

I + (t + t0)(−1 + t2)

(t2 + t2)(t20 + t2)

γ 0 +

+ (t + t0)(t2 + 1)

(t2 + t2)(t20 + t2)

γ 0γ

]+

+ �−n+3

[− −1 + t0t

(t2 + t2)(t20 + t2)

I + t − t0

(t2 + t2)(t20 + t2)

γ 0 −

− 1 + t0t

(t2 + t2)(t20 + t2)

γ − t + t0

(t2 + t2)(t20 + t2)

γ 0γ

].

Hence, the above operator is Huygens in dimension 5 = 2 + 3, in agreement withTheorem 6.

6. Final Remarks

We conclude with a short overview of the main ideas presented in this article aswell as some suggestions for further research. Our main result was the fact that thefields (q, r) constructed from certain explicit polynomial Hirota variables produceDirac operators of the form

D = �∂ − 12(q + r)I + 1

2(q − r)γ (50)


that satisfy Huygens’ principle. Certain reductions of the above families by settingd = 2j + 1 and t2 = t4 = t6 = · · · = 0 in Equation (27) lead to AKNS operatorsassociated to the mKdV hierarchy. In this case, we obtain as a corollary our earlierresult in [11]. This was obtained taking into account that in this case, q = r , and(in the Pauli–Dirac representation of Dirac matrices, where γ 0 = diag[1, −1])

(�∂ − q)(�∂ + q) =[

� − uk+1(x0) 0

0 � − uk(x0)

]. (51)

Furthermore, the potentials in (51) are precisely the ones associated to the (k+1)stand kth Adler–Moser polynomials. In particular, we would get, without much addi-tional effort an independent proof of the Lagnese–Stellmacher result that does notrely on Darboux transformations. It is important, however, to stress that the core ofour proof is to show that Schlesinger transformation is a particular case of Huygenspreserving transformations for Dirac operators (in the sense of Theorem 5), whileLagnese and Stellmacher proved that Darboux transformations are nontrivial trans-formations for D’Alembert operators. So, in our work a Schlesinger transformationperforms a task completely different from Darboux transformations in the previouswork of Lagnese and Stellmacher.

The work of Lagnese and Stellmacher not only found examples of Huygens’type operators but also provided a full classification of Huygens’ preserving per-turbations to the D’Alembertian of the form � + u(x0). The question of whetherour results fully characterize Huygensian perturbations of the form (49) is stillopen. In a forthcoming publication, we show that this question is related to theclassification of all stationary rational solutions of the AKNS hierarchy.

We cannot close this article without a few words on a seemingly unrelatedconcept, namely the bispectral property, which was motivated by problems insignal processing and tomography [20, 21, 51]. An ordinary differential operatorL = L(x, ∂x) is called bispectral if there exists a family of eigenfunctions �(x, k)

of L that also satisfies a (positive order) differential equation in the spectral variablek of the form

B(k, ∂k)� = �(x)�, (52)

where B is independent of x, and �(x) is independent of k. In other words, wehave eigenfunctions that are simultaneously solutions of spectral problems in the‘physical’ and the ‘spectral’ variables.

The connection between bispectrality and integrable systems was first eviden-ced in the seminal work [15] where, besides the characterization of bispectralSchrödinger operators, it was shown that the rational solutions of the KdV hier-archy lead to bispectral potentials for such operators. It was furthered in differentdirections in subsequent works [44, 45, 47–50, 52]. The relation between bispec-trality and Huygens principle for wave operators was highlighted in [8]. See [6, 24]and references therein.


One of the consequences of the present work is another link in the chain of un-expected connections among bispectrality, integrable systems, and Huygens princi-ple. Indeed, it was shown in [50] that the rational solutions of the AKNS hierarchy(which are under consideration in the present work) lead to bispectral AKNS op-erators. The upshot being that the Schur functions discussed in the present workmanifest themselves in three (apparently) distant subjects in a rather surprisingway.

Since Dirac’s equation corresponds to physical systems with spin 1/2, onenatural avenue ushered by the present work is the study of Huygens property forhigher-spin physical systems. As natural candidates we mention Maxwell’s equa-tions (spin 1), Rarita–Schwinger’s equation (spin 3/2), and linearized Einstein’sequation (spin 2). Although there are already studies considering Huygens’ prop-erty for such systems [5], to our knowledge there are no general results connectinghigher spin systems to integrable systems as the one we present here for Dirac inconnection with the rational solutions of the AKNS hierarchy.

Another natural continuation of the present work would be the study of thesymmetry group of Dirac operators of Huygens type. In the wave operator case,Berest analyzed in [8] the group of extended symmetries that preserve Huygensproperty. He showed that the generators of such extended symmetries have a Vi-rasoro type structure and that the KdV flows naturally preserve Huygens principleat the level of the potential u(x0). As it turns out the rational potentials obtainedby Lagnese and Stellmacher belong to the orbit of such flows. We conjecture thatthe rational fields (q, r) presented in this article belong to the orbit of iso-Huygensdeformations for Dirac operators.

Finally, yet another natural follow-up of the present article would be the searchfor Dirac operators of Huygens type depending on more than two functions. In-deed, the representation of Dirac operators leaves room to many other functionsbesides q and r since it involves matrices of size 2(n+1)/2 × 2(n+1)/2, where n + 1is the dimension of spacetime. It would be plausible that such functions are partof a more complex perturbation of the free Dirac operator in an even-dimensionalspacetime.

Acknowledgements

JPZ was supported by CNPq grant number 30.1003 and by PRONEX grant number76.97.1008-00. FACCC was supported by CNPq through grant 142.917/1994-0 andby CAPES. The completion of this work was achieved during the participation atthe 2001 MGSS whose support through NSF grant DMS97-09320 is gratefullyacknowledged. We also thank an anonymous referee for many suggestions andcorrections.


References1. Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H.: The inverse scattering transform-

Fourier analysis for nonlinear problems, Stud. Appl. Math. 53(4) (1974), 249–315.2. Adler, M. and Moser, J.: On a class of polynomials connected with the Korteweg–de Vries

equation, Comm. Math. Phys. 61(1) (1978), 1–30.3. Airault, H., McKean, H. P. and Moser, J.: Rational and elliptic solutions of the Korteweg–

de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30(1) (1977),95–148.

4. Baum, H.: The Dirac operator on Lorentzian spin manifolds and the Huygens property,J. Geom. Phys. 23 (1997), 42–64.

5. Belger, M., Schimming, R. and Wünsch, V.: A survey on Huygens’ principle, Z. Anal.Anwendungen 16(1) (1997), 9–36. Dedicated to the memory of Paul Günther.

6. Berest, Yu. Yu. and Veselov, A. P.: The Huygens principle and integrability, Uspekhi Mat. Nauk49(6(300)) (1994), 7–78.

7. Berest, Yu.: Hierarchies of Huygens’ operators and Hadamard’s conjecture, Acta Appl. Math.53(2) (1998), 125–185.

8. Berest, Yu. Yu.: Deformations preserving Huygens’ principle, J. Math. Phys. 35(8) (1994),4041–4056.

9. Burchnall, J. and Chaudy, T.: A set of differential equations which can be solved bypolynomials, Proc. London Math. Soc. 30 (1929), 401–414.

10. Calogero, F. and Degasperis, A.: Spectral Transform and Solitons, Vol. I, North-Holland Pub-lishing Co., Amsterdam, 1982. Tools to solve and investigate nonlinear evolution equations,Lecture Notes in Computer Science 144.

11. Chalub, F. A. C. C. and Zubelli, J. P.: On Huygens’ principle for Dirac operators and nonlinearevolution equations, J. Nonlin. Math. Phys. 8(Supplement) (2001), 62–68.

12. Chalub, F. A. C. C. and Zubelli, J. P.: Matrix bispectrality and Huygens’ principle for Diracoperators, In: Contemp. Math. 362, Amer. Math. Soc., Providence, RI, 2004.

13. Courant, R. and Hilbert, D.: Methods of Mathematical Physics, Vol. II, Partial DifferentialEquations, Reprint of the 1962 original, Wiley, New York, 1989.

14. Dirac, P. A. M.: The quantum theory of the electron, Proc. Roy. Soc. London A 117 (1928),610–624.

15. Duistermaat, J. J. and Grünbaum, F. A.: Differential equations in the spectral parameter, Comm.Math. Phys. 103(2) (1986), 177–240.

16. Falqui, G., Magri, F., Pedroni, M. and Zubelli, J. P.: An elementary approach to the polynomialτ -functions of the KP-hierarchy, Teoret. Mat. Fiz. 122(1) (2000), 23–36.

17. Flaschka, H.: Construction of conservation laws for Lax equations. Comments on a paper ‘Ontwo constructions of conservation laws for Lax equations’, Quart. J. Math. Oxford Ser. (2)34(133) (1983), 61–65.

18. Folland, G. B.: Fundamental solutions for the wave operator, Exposition. Math. 15(1) (1997),25–52.

19. Gilbert, J. E. and Murray, M. A. M.: Clifford Algebras and Dirac Operators in HarmonicAnalysis, Cambridge University Press, Cambridge, 1991.

20. Grünbaum, F. A.: Band and time limiting, recursion relations, and some nonlinear evolutionequations, In: W. Schempp, R. Askey and T. Koorwinder (eds), Special Functions: GroupTheoretical Aspects and Applications, Reidel, Dordrecht, 1984, pp. 271–286.

21. Grünbaum, F. A.: Some bispectral musings, In: The Bispectral Problem (Montreal, PQ, 1997),Amer. Math. Soc., Providence, RI, 1998, pp. 31–45.

22. Günther, P.: Huygens’ Principle and Hyperbolic Equations, Academic Press, Boston, MA,1988. With appendices by V. Wünsch.

23. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations, DoverPublications, New York, 1953.


24. Harnad, J. and Kasman, A.: The bispectral problem, In: Proc. CRM Workshop, Montreal,Canada, March 1997.

25. Hirota, R.: Direct methods in soliton theory, In: R. Bullough and P. Caudrey (eds), Topics inCurrent Physics, Springer, New York, 1980, pp. 157–175.

26. Ibragimov, N. H.: Huygens’ principle, In: Certain Problems of Mathematics and Mechanics(on the occasion of the seventieth birthday of M. A. Lavrent’ev; in Russian), Nauka, Leningrad,1970, pp. 159–170.

27. Ibragimov, N. Kh. and Oganesyan, A. O.: Hierarchy of Huygens equations in spaces with anontrivial conformal group, Uspekhi Mat. Nauk, 46(3(279)) (1991), 111–146, 239.

28. Jimbo, M. and Miwa, T.: Monodromy preserving deformation of linear ordinary differentialequations with rational coefficients. II, Phys. D 2(3) (1981), 407–448.

29. Jimbo, M. and Miwa, T.: Monodromy preserving deformation of linear ordinary differentialequations with rational coefficients. III, Phys. D 4(1) (1981/1982), 26–46.

30. Kricever, I. M.: On the rational solutions of the Zaharov–Šabat equations and completely in-tegrable systems of N particles on the line, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst.Steklov. (LOMI) 84 (1979), 117–130, 312, 318. Boundary value problems of mathematicalphysics and related questions in the theory of functions, 11.

31. Lagnese, J. E. and Stellmacher, K. L.: A method of generating classes of Huygens’ operators,J. Math. Mech. 17 (1967), 461–472.

32. Marchuk, N. G.: Dirac γ -equation, classical gauge fields and Clifford algebra, Adv. Appl.Clifford Algebras 8(1) (1998), 181–225.

33. Mumford, D.: Tata Lectures on Theta. I. With the assistance of C. Musili, M. Nori, E. Previatoand M. Stillman, Birkhäuser, Boston, MA, 1983.

34. Newell, A. C.: Solitons in Mathematics and Physics, Society for Industrial and AppliedMathematics (SIAM), Philadelphia, PA, 1985.

35. Sachs, R. L.: On the integrable variant of the Boussinesq system: Painlevé property, rationalsolutions, a related many-body system, and equivalence with the AKNS hierarchy, Phys. D30(1–2) (1988), 1–27.

36. Sakurai, J. J.: Advanced Quantum Mechanics, Addison-Wesley, Englewood Cliffs, 1967.37. Sato, M. and Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grass-

mann manifold, In: Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982),North-Holland, Amsterdam, 1983, pp. 259–271.

38. Schimming, R.: Korteweg–de Vries Hierarchie und Huygensschess Prinzip, In: DresdenerSeminar für Theoretische Phyzik, Sitzungsberichte Nr. 26, 1986.

39. Schimming, R.: Laplace-like linear differential operators with a logarithm-free elementarysolution, Math. Nachr. 148 (1990), 145–174.

40. Stellmacher, K. L.: Ein Beispiel einer Huyghensschen Differentialgleichung, Nachr. Akad.Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys. Chem. Abt. 1953 (1953), 133–138.

41. Stellmacher, K. L.: Eine Klasse Huyghenscher Differentialgleichungen und ihre Integration,Math. Ann. 130 (1955), 219–233.

42. Thaller, B.: The Dirac Equation, Springer-Verlag, Berlin, 1992.43. Wetterich, C.: Massless spinors in more than four dimensions, Nuclear Phys. B 211(1) (1983),

177–188.44. Wilson, G.: Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442

(1993), 177–204.45. Wilson, G.: Collisions of Calogero–Moser particles and an adelic Grassmannian (with an

appendix by I. G. Macdonald), Invent. Math. 133(1) (1998), 1–41.46. Zakharov, V. E. and Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-

dimensional self-modulation of waves in nonlinear media, Zh. Èksper. Teoret. Fiz. 61(1) (1971)118–134.


47. Zubelli, J. P.: Differential equations in the spectral parameter for matrix differential operators,Phys. D 43 (1990), 269–287.

48. Zubelli, J. P.: On the polynomial τ -functions for the KP hierarchy and the bispectral property,Lett. Math. Phys. 24 (1992), 41–48.

49. Zubelli, J. P. and Magri, F.: Differential equations in the spectral parameter, Darboux transfor-mations, and a hierarchy of master symmetries for KdV, Comm. Math. Phys. 141(2) (1991),329–351.

50. Zubelli, J. P.: Rational solutions of nonlinear evolution equations, vertex operators, andbispectrality, J. Differential Equations 97(1) (1992), 71–98.

51. Zubelli, J. P.: Topics on Wave Propagation and Huygens’ Principle, Instituto de MatemáticaPura e Aplicada (IMPA), Rio de Janeiro, 1997.

52. Zubelli, J. P. and Valerio Silva, D. S.: Rational solutions of the master symmetries of the KdVequation, Comm. Math. Phys. 211(1) (2000), 85–109.


Toward Verification of the Riemann Hypothesis:Application of the Li Criterion

MARK W. COFFEYDepartment of Physics, Colorado School of Mines, Golden, CO 80401, U.S.A.e-mail: [email protected]

(Received: 12 January 2004; accepted: 19 May 2005)

Abstract. We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon aseries representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemannxi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums andthe sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of asequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative ofthe zeta function about s = 1 and appear to have remarkable characteristics. On our conjecture, notonly does the Riemann hypothesis follow, but an inequality governing the values λn and inequalitiesfor the sums of reciprocal powers of the nontrivial zeros of the zeta function.

Mathematics Subject Classification (2000): 11M26.

Key words: Riemann zeta function, Riemann xi function, logarithmic derivatives, Riemann hy-pothesis, Li criterion, Stieltjes constants, Laurent expansion, L-function, Dirichlet series, HeckeL-function, Dedekind zeta function, extended Riemann hypothesis.

1. Introduction

In this paper, we reduce the verification of the Riemann hypothesis to a conjectureconcerning the behavior of certain coefficients ηk which appear in the Laurentexpansion of the logarithmic derivative of the zeta function about s = 1. Moreover,should the conjectured property hold, we would then derive a result stronger thanthe Riemann hypothesis itself. Namely, we would have obtained an explicit lowerbound for a sequence {λj } of certain logarithmic derivatives of the xi functionevaluated at unit argument. Since the coefficients ηj can be written as particularlimits involving the von Mangoldt function �, these quantities seem to encapsulateboth number theoretic and analytic information. Such information is expected to beencountered in any rigorous denial or verification of the Riemann hypothesis.

We first present an overview of our approach. We shall use the Li equivalencefor the Riemann hypothesis to hold [32]. This equivalence results from a necessaryand sufficient condition that the logarithmic derivative of the function ξ [1/(1 − z)]be analytic in the unit disk. The function ξ is determined from the Riemann zetafunction ζ by way of the relation ξ(s) = (s/2)(s − 1)π−s/2�(s/2)ζ(s), where

212 MARK W. COFFEY

� is the Gamma function [16, 26, 28, 45, 46]. Then the xi function satisfies thefunctional equation ξ(s) = ξ(1−s). The Li equivalence states that a necessary andsufficient condition for the nontrivial zeros of the Riemann zeta function to lie onthe critical line Re s = 1/2 is that a sequence of real numbers λk is nonnegativefor every integer k. In this paper, we investigate the behavior of this sequence,based upon a series representation previously derived [12]. Of this representation,we are able to bound one finite sum and make progress in bounding the other. Wefind that indeed the sequence {λk} has only nonnegative numbers, subject to theconjectured properties of the sequence ηj . Among other connections, the ηj canbe readily related to the Stieltjes constants γk. This gives one of several avenuesfor further investigation of the ηj expansion coefficients. This paper includes aseries of Appendices A–L which contain various extensions of our summationestimations, occasional alternative proofs, series representations analogous to thatfor λn relevant to other L-functions, tabulated numerical values, derivative relationsof the Riemann zeta function, alternative expansions of the logarithmic derivativeof the Riemann zeta and xi functions, remarks on the integer-order derivatives ofthe Dedekind zeta and xi functions, and other reference material.

We stress that we do not just conjecture as to the nature of the ηj ’s, but pro-vide a perhaps strong plausibility argument in favor. In addition, current numericalevidence [14, 36] seems to fully support our conjecture.

The sequence {λn}∞n=1 is defined by

λn = 1

(n − 1)!dn

dsn[sn−1 ln ξ(s)]s=1. (1)

Then Li’s criterion for the Riemann hypothesis to hold is that all {λn}∞n=1 are non-negative [32]. We note that Li’s convention for the xi function has a factor of twodifference: ξLi(s) = 2ξ(s), although this is immaterial in logarithmic derivativessuch as λj . We also have

λn = (−1)n

(n − 1)!dn

dsn[(1 − s)n−1 ln ξ(s)]s=0, (2)

and ξ(0) = −ζ(0) = 1/2. (Hence ξLi(0) = 1.) The approximate numerical valuesfor the first few λj ’s are: λ1 � 0.0230957, λ2 � 0.0923457, and λ3 � 0.207639(see Appendix D). In fact, we have λ1 = −B, where B ≡ ln 2 + (1/2) ln π − 1 −γ /2 � −0.0230957, and γ � 0.5772156649 is the Euler constant. This followsfrom the logarithmic derivative [26]

ξ ′(s)ξ(s)

= B +∑

ρ

(1

s − ρ+ 1

ρ

), (3)

TOWARD VERIFICATION OF THE RIEMANN HYPOTHESIS 213

where ρ runs over all the nontrivial zeros of the zeta function. Thus ξ ′(s)/ξ(s) =∑ρ(s − ρ)−1, which is consistent with ξLi(s) = ∏

ρ(1 − s/ρ). In general, the λj ’sare connected to sums over the nontrivial zeros of ζ(s) by way of [32]

λn =∑

ρ

[1 −

(1 − 1

ρ

)n]. (4)

By using the Laurent expansion of the zeta function about s = 1,

ζ(s) = 1

s − 1+

∞∑

n=0

(−1)n

n! γn(s − 1)n, (5)

where the γk are the Stieltjes constants [8, 22, 24–26, 29, 38], it is possible towrite a closed form for the λj ’s. The Stieltjes constants can be evaluated from theexpression

γk = limN→∞

(N∑

m=1

1

mlnk m − lnk+1 N

k + 1

), (6)

and several other forms have been given [8, 22, 24, 25, 29, 38]. For instance, wehave

λ2 = 1 + γ − γ 2 + π2

8− 2 ln 2 − ln π − 2γ1, (7)

and

λ3 = 12 [2 + 3

4π2 − 6 ln 2 − 3 ln π − 12γ1 + γ [3 + 2(γ − 3)γ + 6γ1] +

+ 3γ2 − 74ζ(3)]. (8)

It is not difficult to determine that λj contains the term −(−1)j [j/(j − 1)!]γj−1.The successive λj ’s can be related in several different ways, including simply

λn+1 = λn + 1

n![

dn

dsnsn ξ ′(s)

ξ(s)

]

s=1

. (9)

Alternatively, in the particular case of λ2, one can write λ2 = 2λ1 − λ21 +

lims→1 sξ ′′(s)/ξ(s). Elsewhere, we have very recently obtained a general explicitrelation for λk in terms of the Stieltjes constants [13].

2. Alternative Representation of Li’s λj ’s

Of particular importance for the purposes of this paper is an alternative formula forthe particular sequence of logarithmic derivatives of the Riemann xi function givenin Equation (1). Due to the centrality of this result, we briefly review the derivation[12] of the following representation,

214 MARK W. COFFEY

THEOREM 1.

λn = −n∑

m=1

(n

m

)ηm−1 +

n∑

m=2

(−1)m

(n

m

)(1 − 2−m)ζ(m) +

+ 1 − n

2(γ + ln π + 2 ln 2), (10)

where the constants ηj can be written as

ηk = (−1)k

k! limN→∞

(N∑

m=1

1

m�(m) lnk m − lnk+1 N

k + 1

), (11)

and � is the von Mangoldt function [16, 26–28, 45, 46].

From the expansion around s = 1 of the logarithmic derivative of the zetafunction,

ζ ′(s)ζ(s)

= −(s − 1)−1 −∞∑

p=0

ηp(s − 1)p, (12)

we have

ln ζ(s) = − ln(s − 1) −∞∑

p=1

ηp−1

p(s − 1)p, (13)

giving

ln ξ(s) = − ln 2 + ln s − s

2ln π + ln �

(s

2

)−

∞∑

p=1

ηp−1

p(s − 1)p. (14)

The radius of convergence of the expansion (12) is 3, as the first singularity en-countered is the trivial zero of ζ(s) at s = −2. We then evaluate

dn

dsn[sn−1 ln ξ(s)]s=1

= (n − 1)!n−1∑

j=0

(n

j

)1

(n − j − 1)![

dn−j

dsn−jln ξ(s)

]

s=1

, (15)

using in particular the special values ψ(1/2) = −γ − 2 ln 2 and ψ(n)(1/2) =(−1)n+1n!(2n+1 −1)ζ(n+1) for n � 1, where ψ = �′/� is the digamma functionand ψ(j) is the polygamma function. Finally, the sum in Equation (15) over j canbe converted to a sum over m = n−j and the simple result − ∑n

m=1(−1)m(

n

m

) = 1used, yielding Equation (10).

The Laurent expansion of ζ ′/ζ with the form of the constants ηj in Equa-tion (11) can be developed by applying Theorem 1 of [27]. In this case, the counting


function A(x) = − ∑n�x �(n) = −ψ(x), where ψ is the Chebyshev function,

and the error term is given by u(x) = x − ψ(x). Equation (10) has been derivedalternatively in [7] by a method connected with A. Weil’s explicit formula. Ourapproach is independent, and we believe, more direct. In Appendix G we presentan extension of Theorem 1 which accounts explicitly for the presence of the firstsix trivial zeros of the zeta function.

3. Estimation of Sums

We characterize each of the two summation terms on the right side of Equation (10)in turn. Before this discussion, we emphasize that the first few λj ’s may be explic-itly written, as indicated above, and directly verified to be positive. In addition, wenote that apparently it is already known [6] that λn � 0 for all n � 2.975 · · ·×1017.

The sum

S1(n) ≡n∑

m=2

(−1)m

(n

m

)(1 − 2−m)ζ(m) (16)

may be written in several equivalent ways, including

S1(n) =∞∑

k=0

n∑

m=2

(−1)m

(n

m

)1

(2k + 1)m

=∞∑

k=0

[n

2k + 1− 1 + 2nkn

(2k + 1)n

]. (17)

Equation (17) results from the use of (e.g., [1])

(1 − 2−n)ζ(n) =∞∑

k=0

(2k + 1)−n, n � 2, (18)

and interchange of the order of the two summations in Equation (16). It appears tobe equally profitable to write S1 as

S1(n) =∞∑

k=1

n∑

m=2

(−1)m

(n

m

)(1 − 2−m)

1

km

=∞∑

k=1

[n

2k+

(1 − 1

k

)n

−(

1 − 1

2k

)n]. (19)

By using integrals estimating these forms, we obtain

THEOREM 2.

S1(n) � n

2ln n + (γ − 1)

n

2+ 1

2 . (20)

216 MARK W. COFFEY

Remark. By inserting an integral representation for ζ into Equation (16), it ispossible to obtain the sum S1 in the form

S1(n) = n

2

∫ ∞

0

[1 + 1F1

(1 − n; 2; t

2

)− 2 1F1(1 − n; 2; t)

]dt

(et − 1),

where 1F1 is the Kummer confluent hypergeometric function. In this equation,n 1F1(1 − n; 2;w) = L1

n−1(w), where Lαn is an associated Laguerre polynomial.

Proof of Theorem 2. We proceed on the basis of comparison to the form ofEquation (17), leaving the comparison to Equation (19) to Appendix A. In addition,we relegate to Appendix L the application of Euler–Maclaurin summation to S1(n),which gives improved estimates of the linear term.

By making a change of variable in the integral

I1(n) =∫ ∞

0

[n

2k + 1− 1 + 2nkn

(2k + 1)n

]dk, (21)

we obtain

I1(n) = 1

2

∫ ∞

1

[n

x− 1 + (x − 1)n

xn

]dx. (22)

Another change of variable and an integration by parts yields

I1(n) = n

2

∫ 1

0[1 − (1 − t)n−1]dt

t− 1

2(n − 1). (23)

Evaluation of the integration [20] then gives

I1(n) = n

2[ψ(n) + γ − 1] + 1

2 . (24)

Since the integrand of Equation (21) is monotonically decreasing with k, the in-equality (20) readily follows. �COROLLARY. With the other sum term in Equation (10) denoted as S2, we havealready obtained the inequality λn � (n/2) ln n− (n/2)(1 + ln π + 2 ln 2)+ 3/2 −|S2|. Similarly, it is possible to readily obtain an upper bound for S1 and λn. Wehave, for instance, S1(n) � (n/2) ln n+ (γ +1)n/2−1/2. It is possible to developimproved coefficients for the linear in n term, giving tighter bounds for S1(n), asshown in Appendix L.

Remarks. (1) Since Li has very recently obtained explicit formulas for bothDirichlet and Hecke L-functions [34], which are analogous to Equation (10), we


may expect our summation estimation techniques to aid in making progress in ver-ifying the extended and generalized Riemann hypotheses. In fact, we may observethat the term τχ(n) defined in Theorem 1 of [34] can be rewritten as

τχ(n) = S1 − n ln 2 if χ(−1) = 1,

= S0 − S1 if χ(−1) = −1,(25)

where S0 is defined in Equation (A15) of Appendix A. The second line of Theo-rem 2 of [34] for λE(n) may be written as

2(1 − 13 ln 2)n + S3 − S0 + 2n + (−1)n − 1, (26)

where S3 is defined in Equation (A18) of Appendix A. Inequality results basedupon the use of Equations (25) and (26) and the estimations of Appendix A aregiven in Appendix E, while estimations of generalized Riemann zeta functionssums are presented in Appendix F.

(2) In [47], the rate of growth of the sum S1(n) is conjectured. An integral ex-pression for the Li constants is written and a saddle point method is applied. How-ever, the dominant O(n ln n) behaviour that we have demonstrated is not rigorouslyobtained.

It is not difficult to show that the following recursion relation exists between theStieltjes constants of Equation (5) and the coefficients ηj . We have η0 = −γ0 =−γ and

ηn = −(−1)n

[(n + 1)

n! γn +n−1∑

k=0

(−1)k−1

(n − k − 1)!ηkγn−k−1

]. (27)

This equation is equivalent to the statement [ζ ′(s)/ζ(s)]ζ(s) = ζ ′(s) and will beapplied later on.

It is also of interest to relate the sequence {ηj } to the sequence {σk} which occursin the expansion

ln ξ(s) = − ln 2 −∞∑

k=1

(−1)k σk

k(s − 1)k. (28)

The ensueing relation figures prominently in our conjecture for the behaviour ofthe coefficients ηj . One reason for the importance of the coefficients σk is theircorrespondence to sums of reciprocal powers of the nontrivial zeros ρ of the ζ

function [31, 39]: σk = ∑ρ ρ−k. From Equation (4) we see that the connection

between the values λn and the sequence {σk} is λn = − ∑nj=1(−1)j

(n

j

)σj . Further

discussion of the σj ’s is presented in Appendix J. We shall next demonstrate.

THEOREM 3.

σk = (−1)kηk−1 − (1 − 2−k)ζ(k) + 1, k � 2. (29)

218 MARK W. COFFEY

Proof. The key to showing that Equation (29) holds is to evaluate the particularsuccessive Riemann zeta function sums

∞∑

n=k

ζ(n) − 1

2n

(n − 1

k − 1

)= (1 − 2−k)ζ(k) − 1, k � 2. (30)

From Equation (14) and the expansion [1]

ln

[s�

(s

2

)]= ln 2 + γ − 1

2+ 1 − γ

2(s − 1) +

+∞∑

n=2

ζ(n) − 1

n2n[1 − (s − 1)]n, (31)

we have

ln ξ(s) = − 12 ln π + γ − 1

2+

∞∑

n=2

ζ(n) − 1

n2n+

+ (1 − s)

[ln π

2− 1

2(γ + 1) +

∞∑

n=2

ζ(n) − 1

2n

]−

−∞∑

p=2

ηp−1

p(s − 1)p +

∞∑

n=2

n∑

j=2

ζ(n) − 1

n2n

(n

j

)(1 − s)j . (32)

Upon comparison with Equation (28), the constant term in this equation (i.e., thecoefficient of the (1 − s)0 term) is readily shown to be zero (or see Appendix B)and the linear term immediately gives

σ1 = − ln π

2+ γ

2+ 1 − ln 2 = λ1. (33)

We then have

−∞∑

p=2

ηp−1

p(s − 1)p +

∞∑

n=2

n∑

j=2

ζ(n) − 1

n2n

(n

j

)(1 − s)j

= −∞∑

k=2

σk

k(1 − s)k. (34)

Upon reordering the two sums we obtain

σk = (−1)kηk−1 −∞∑

n=k

ζ(n) − 1

2n

(n − 1

k − 1

), k � 2. (35)


We now need to demonstrate Equation (30). We first note that

∞∑

n=k

ζ(n) − 1

2n

(n − 1

k − 1

)

=∞∑

�=2

1

(k − 1)!∞∑

n=k

(n − 1)(n − 2) · · · (n − k + 1)

(2�)n(36)

and∞∑

k=j+1

k(k − 1)(k − 2) · · · (k − j)qk−j−1 = (j + 1)!(1 − q)j+2

, |q| < 1. (37)

Equation (37) is easily verified by mathematical induction. With Equation (36),we have proceeded by interchanging two summations. However, we could haveequally well used the integral representation

ζ(s) = 1

�(s)

∫ ∞

0

t s−1

(et − 1)dt, Re s > 1, (38)

and then interchanged the order of summation and integration, which is carried outin Appendix B. By applying Equation (37) to Equation (36) we obtain the succinctresult

∞∑

n=k

ζ(n) − 1

2n

(n − 1

k − 1

)=

∞∑

�=1

1

(2� + 1)k. (39)

If we then invoke Equation (18) we obtain successively Equations (30) and (29).Theorem 3 is proved in another manner in Appendix C. �

Since in Equation (12) ζ ′(s)/ζ(s) + (s − 1)−1 is a transcendental function ana-lytic in the disc |s − 1| < 3, there is an infinite number of ηj ’s which are nonzero.From this equation we have a good deal of information on sums and moments ofthis sequence. For instance, we have

∞∑

p=0

ηp = −[

1 + ζ ′(2)

ζ(2)

]� −0.430039 < 0, (40a)

∞∑

p=0

(−1)pηp = 1 − ln 2π � −0.837877 < 0, (40b)

and more generally, by taking successive derivatives,

∞∑

p=j

p(p − 1) · · · (p − j + 1)ηp = −(−1)j j ! −[ζ ′(s)ζ(s)

](j)∣∣∣∣s=2

. (41)

220 MARK W. COFFEY

Various formulas for zeta function derivatives, and in particular concerning thelogarithmic derivative occurring in this equation, are presented in Appendix H.Furthermore, Equation (13) gives

ln ζ(2) = −∞∑

p=1

ηp−1

p, (42a)

and

ln 2 =∞∑

p=1

(−1)p ηp−1

p. (42b)

By evaluating Equation (12) at s = 1/2 and using the functional equation we obtain

∞∑

p=0

(− 12)

pηp = 2 − 12 [ln π − ψ( 1

4)], (43a)

or∞∑

p=1

(− 12)

pηp = 2 −[

ln π − γ + π

2+ 3 ln 2

]. (43b)

An equation such as (40a) shows that not all the ηj ’s can be positive. In fact, wecan argue that there is an infinite number of ηj ’s which are positive, and an infinitenumber which are negative. This result, together with the fact that the sequence ηj

decreases to zero with j increasing to infinity, as shown by either Equation (40a)or (41), provides a THEOREM 4.

By multiplying Equation (12) by sq with q > −1 and integrating over s from 0to 1 we obtain

∞∑

p=0

(−1)pB(p + 1, q + 1)ηp

= −[ψ(q + 1) + γ ] −∫ 1

0

[sq ζ ′(s)

ζ(s)+ 1

s − 1

]ds, (44)

where B is the beta function.We now characterize the behaviour of the sequence {ηj } as a function of its

index. We have

THEOREM 5.

|ηj−1| � |σj | + 2−j ζ(j), (45)

and


CONJECTURE 1. (i) The sequence {σj } decays with j faster than 1/3j . Then,(ii) for j sufficiently large, the sequence {ηj } alternates in sign and decreases inmagnitude as approximately 1/3j . (iii) For all j � 0, the magnitudes |ηk| satisfy|ηk| � γ 2−k. Furthermore, (iv) the sequence {ηj } alternates in sign for all j � 0.

Remark. Since the radius of convergence of the expansion (12) is 3, we knowthat |ηj | cannot increase faster than 1/3j for sufficiently large j . In fact, the expres-sion ηj � −γ (−1/3)j for all j � 0 is a very good approximation. One may readilyverify this assertion, for instance, with the sums appearing in Equations (40)–(43). In addition, all currently known numerical evidence [36] supports both thisapproximation and the strict sign alternation suggested in the conjecture.

We cannot currently prove all parts of Conjecture 1, but we can offer what webelieve to be a strong plausibility argument. In the course of this argument we doprove Theorem 5. In addition, based upon known properties of σj , we very recentlyhave proved part (iv) on the strict sign alternation of the {ηj } sequence for all valuesof j [13].

(i) The first nontrivial zero ρ1 of the zeta function is known to lie on the criticalline and to have ordinate approximately given by 14.134725142 (e.g., [19, 41]),i.e., ρ1 = 1/2 + iα1. This zero, along with its complex conjugate, dominate thesum σk so that for large k we have σk ∼ α−k

1 , a rate of decrease much faster than3−k. In fact, depending upon whether k is even or odd, and if even, divisible or notby 4, a leading behaviour of σk is given by one of the four forms ±2αk

1/(1/4+α21)

k

(for k even) or ±kαk−11 /(1/4 + α2

1)k (for k odd). In addition, one may argue much

more conservatively with the expression

( 12 − iα1)

k + ( 12 + iα1)

k = 2k∑

j=0

(k

j

)α

j

1

2k−j

1 if j = 4m,0 if j = 4m + 1,

−1 if j = 4m + 2,0 if j = 4m + 3

(46)

by taking the approximate largest value of the binomial coefficient and ignoringthe attenuating effect of the powers of 1/2. This will yield a form of approximatelyσk ∼ (2/

√k)(2/α1)

k, which is still a much faster decrease with k than 1/3k.For both emphasis and clarity, we restate Equation (29) as

ηj−1 = (−1)j [σj + (1 − 2−j )ζ(j) − 1], j � 2. (47)

By appeal to Equation (18) we have

2−j ζ(j) > (1 − 2−j )ζ(j) − 1 =∞∑

k=1

1

(2k + 1)j>

1

3j, j � 2. (48)

Applying the triangle inequality to Equation (47) and using the left inequality in(48) gives Theorem 5.

222 MARK W. COFFEY

Continuing with the argument for part (ii) of Conjecture 1, by part (i),(1−2−j )ζ(j)−1 > 0 then dominates in the brackets in Equation (47) and the signalternation of the sequence {ηj } then follows for sufficiently large j .

(iii) With the aid of the recursion relation (27) or by several other means, itis possible to calculate ηk for any desired initial set k = 1, . . . , k0 and directlyverify their sign alternation and the stated inequality (e.g., as in Appendix D). Forlarger values of k, this inequality may hold due to the left inequality in (48) whencombined with Equation (47).

Remark. It is not essential to our main purpose here, but we may comment onthe sign pattern of the {σk} sequence. The initial, and in a sense typical, sign patternis simply −−++−−· · ·, with σ1 > 0. Initially, as k takes on the respective values4m, 4m + 1, 4m + 2, 4m + 3, where m is a positive integer, the sign of σk is givenby +, +, −, −. This pattern continues to the point where k(k − 1) > 8α2

1. Thisexplains why σ46 > 0 rather than σ46 < 0. Similar considerations apply for largervalues of k in this sequence.

Mainly for reference purposes, we will now indicate other possible uses of therecursion relation (27). It has been proved that [5]

|γn| � [3 + (−1)n](n − 1)!πn

, n � 1, (49)

which has been improved to [39]

|γn| � [3 + (−1)n](2n)!nn+1(2π)n

, n � 1. (50)

As an illustration of the use of such results, the combination of Equations (27) and(49) gives

LEMMA 1.

|ηn| � 1

πn

{(n + 1)

n[3 + (−1)n] +

+n−1∑

k=1

[3 + (−1)n−k]πk+1

(n − k)|ηk−1|

}+ γ 2, n � 1. (51)

Similarly, the inequality (50) may be applied to Equation (27), permitting, forexample, inductive arguments on |ηn|, but we have already deduced Theorem 4.

We are now in position to estimate the sum

S2(n) ≡ −n∑

m=1

(n

m

)ηm−1, (52)


of Equation (10). On Conjecture 1, we have that the ηj ’s decrease in magnitudewith j and always alternate in sign. This means that, similar to the behaviour of thesum S1, there is a near exponential amount of cancellation in the sum S2. Indeed,we have

CONJECTURE 2.

|S2| � 3γ + C2n1/2+ε, (53)

where C2 is a positive constant and ε is positive and arbitrarily small.

This conjecture is partially motivated by our discussion elsewhere [13] of thepossible connection between the Stieltjes and Li constants and Brownianmotion [6].

The combination of Equation (10) and the inequalities (20) and (53) results in

CONJECTURE 3.

λn � n

2ln n − n

2(1 + ln π + 2 ln 2) + 3

2 − 3γ − C2n1/2+ε, (54)

where the approximate numerical value of the coefficient 1 + ln π + 2 ln 2 is 3.53.

This estimation would show, in the absence of the last two terms on the rightside, that already for values of n exceeding 34, we would be ensured that all λn’sare nonnegative. In addition, as previously mentioned, for smaller values of n onehas only to directly calculate these particular logarithmic derivatives, from eitherEquations (1), (2), or Equation (10) itself and verify the nonnegativeness of theλj ’s (Appendix D).

4. Summary and Brief Discussion

Our program for verification of the celebrated Riemann hypothesis should nowbe clear. We have invoked the Li equivalence [32], wherein it is necessary todemonstrate the nonnegativity of the sequence {λn}∞n=1. Our starting point hasbeen the reformulation of the definition (1) as the series representation (10), λn =S1 + S2 + 1 − n(γ + ln π + 2 ln 2)/2. Some attention to the sums S1 and S2

[Equations (16) and (52)] has been required because they exhibit the phenomenonof exponential cancellation. That is, the binomial coefficient

(n

m

)within the two

summands can take on values approaching 2n/√

n. The strict sign alternation in thesummands of S1 and S2 is critical. For the sum S1 the sign alternation is explicit,while for S2 it has to be deduced [13], as in Conjecture 1. It should be mentionedthat the line of reasoning suggested here for a complete proof of the Riemannhypothesis does not require strict sign alternation of the sequence {ηj } – it is justthat this result could make the estimation of the sum S2 much easier. In fact, veryrecently we have proved the strict sign alternation of this sequence [13]. What is

224 MARK W. COFFEY

next required is an argument that correlates the magnitude of say ηj to ηj+1. Aspreviously mentioned, current numerical evidence [14, 36] seems to support Con-jecture 1 for the behaviour of the {ηj } and {σk} sequences. Certainly both furthercomputation and analysis appear to be in order.

In contrast to the Stieltjes constants γj , whose sign pattern is not so easy todiscern, that of the two sequences {ηk}∞k=0 and {λk}∞k=1 seems to be remarkablysimple. Upon Conjecture 1 and Equations (29) and (48), our approach would alsoyield various inequalities for the sums σk of reciprocal powers of the nontrivialzeros of the zeta function.

To put it very mildly, many implications could follow from the results presented.We may stress that the conjectured behaviour of the sequence {ηj } has multipleimplications for the von Mangoldt and Chebyshev functions, among many others.

We may additionally stress, that should the claims of the propositions and con-jectures herein indeed be valid, we would have not only verified the Riemann hy-pothesis but produced yet a stronger result. Namely, we would have developed theinequality (54) for the sequence {λn}. Moreover, it seems that an asymptotic versionof inequality (54) will suffice to verify the Riemann hypothesis since evidently [6]so many λj ’s are already known to be nonnegative.

The approach of this paper suggests that verification of the Riemann hypothesismay be possible within analysis. Indeed, our approach may be amenable to con-fronting the generalized Riemann hypothesis. Some of our sum estimations carryover immediately to the explicit formulas for λχ(n) and λE(n) in Theorems 1 and 2respectively in [34]. In turn, one is left with estimating sums which contain the vonMangoldt function, reciprocal powers of k, powers of ln k, and Dirichlet characters.To us, this appears to be a realistic approach to the generalized Riemann hypothesisfor Dirichlet and Hecke L-functions and the Dedekind zeta function. We include inAppendix E example results along this line of investigation.

Acknowledgement

I thank J. C. Lagarias for useful discussion concerning a fall 2003 version of thismanuscript.

Appendix A: Estimation of the Sum S1 of Equation (16) and of Other Sums

Here we present the derivation of the inequality (20) for the sum S1 of Equa-tion (16), based upon the form (19). By making a change of variable in the integral

I1(n) ≡∫ ∞

1

[n

2k+

(1 − 1

k

)n

−(

1 − 1

2k

)n]dk, (A1)

we obtain

I1(n) =∫ 1

0

[n

2y + (1 − y)n −

(1 − y

2

)n]dy

y2. (A2)


We then evaluate this integral with an integration by parts, resulting in

I1(n) = n

2

{ψ(n) + γ − 1 +

+∫ 1

0

[(1 − y

2

)n−1

− (1 − y)n−1

]dy

y

}+ 2−n, (A3)

where ψ is the digamma function. Since the remaining integral is nonsingular andO(1) the inequality (20) follows. However, we may continue much further.

By adding and subtracting 1 in the integrand in Equation (A3), we may write

I1(n) = n

2[ψ(n) + γ − 1] +

+ n

2

∫ 1

0

[(1 − y

2

)n−1

− 1

]dy

y+ n

2[ψ(n) + γ ] + 2−n, (A4)

where

∫ 1

0

[(1 − y

2

)n−1

− 1

]dy

y

= −ψ(n) − γ −∫ 1

1/2

(1 − w)n−1

wdw + ln 2. (A5)

The integral on the right side of Equation (A5) may be evaluated in multiple ways.A first method is to use the Gauss hypergeometric function 2F1:

∫ 1

1/2

(1 − w)n−1

wdw = 1

2n−1

∫ 1

0

(1 − x)n−1

1 + xdx

= 1

n2n−1 2F1(1, 1; n + 1;−1), (A6)

where

2F1(1, 1; n + 1;−1) =∞∑

j=0

(−1)j j !(n + 1)j

= n!∞∑

j=0

(−1)j j !(j + n)! , (A7)

and (.)k is the Pochhammer symbol. By using a partial fractional decomposition ofthe summand in Equation (A7), it is possible to show that the sum contains a termn2n−1 ln 2, leading to a possibly new reduction of the particular 2F1 at minus unitargument. By applying the well known formula (e.g., [21])

N !x(x + 1) · · · (x + N)

=N∑

k=0

(N

k

)(−1)k

x + k, (A8)

226 MARK W. COFFEY

whose right side is simply (−1)N�N(1/x), where � is the difference operator,�f (x) = f (x + 1) − f (x), we obtain

2F1(1, 1; n + 1;−1) = n

n−1∑

k=0

(−1)k

(n − 1

k

) ∞∑

j=0

(−1)j

j + k + 1

= n

[2n−1 ln 2 −

n−1∑

k=0

(n − 1

k

) k−1∑

j=0

(−1)j

j + 1

]. (A9)

Next, we proceed alternatively. By way of binomial expansion in the integrandof the second integral in Equation (A6), we have

∫ 1

0

(1 − x)n−1

1 + xdx =

n−1∑

j=0

(−1)j

(n − 1

j

)β(j + 1), (A10)

where [20]

β(j + 1) = (−1)j ln 2 +j∑

k=1

(−1)k+j

k, (A11)

giving

∫ 1

1/2

(1 − w)n−1

wdw = ln 2 + 1

2n−1

n−1∑

j=0

(n − 1

j

) j∑

k=1

(−1)k

k. (A12)

This result is in agreement with Equations (A6) and (A9). The sum in Equa-tion (A12) is easily estimated, leading to

−1 + 1

2n� n

2

∫ 1

1/2

(1 − w)n−1

wdw � 1 − 1

2n. (A13)

Using Equations (A4), (A5), (A12), and (A13) gives

I1(n) � n

2[ψ(n) + γ + ln 2 − 1] − 1 + 21−n. (A14)

The approach followed here can be used to estimate many other Riemann zetafunction sums of interest. For instance, we have

S0(n) ≡n∑

m=2

(−1)m

(n

m

)ζ(m) =

∞∑

k=1

[n

k− 1 +

(1 − 1

k

)n]. (A15)

Then one can determine

I0(n) =∫ ∞

1

[n

k− 1 +

(1 − 1

k

)n]dk =

n∑

j=2

(n

j

)(−1)j

(j − 1), (A16)


obtained by binomial expansion and term-by-term integration. On the other hand,by using

I0(n) =∫ 1

0[ny − 1 + (1 − y)n]dy

y2, (A17)

and an integration by parts, we have S0 � I0 = n[ψ(n) + γ − 1] + 1. Since thedigamma function satisfies [1] ψ(x) = ln x −1/2x −1/12x2 +O(x−4) as x → ∞,the inequality S0(n) � n(ln n + γ − 1) + 1 follows for n � 2.

Additionally, we may estimate

S3(n) ≡n∑

j=2

(−1)j

(n

j

)2j ζ(j). (A18)

Then we have the comparison integral

I3(n) =∫ ∞

1

[2n

k− 1 +

(1 − 2

k

)n]dk

=∫ 1

0[2ny − 1 + (1 − 2y)n]dy

y2. (A19)

An integration by parts gives

I3(n) = 2n

∫ 1

0[1 − (1 − 2y)n−1]dy

y− 2n + 1 − (−1)n, (A20)

and then we have

I3 = (−1)nn

{ψ

(1 − n

2

)− ψ

(n

2

)+ 2(−1)n[γ + ln 2 + ψ(n)] +

+ π tan

(nπ

2

)}− 2n + 1 − (−1)n. (A21)

One may note that the last three terms on the right side of Equation (A21) enterwith opposite signs from Equation (26). The form of I3 in Equation (A21) maybe further rewritten with the use of the relation [20] ψ[(1 − n)/2] =ψ[(n + 1)/2] − π tan(nπ/2), together with the doubling formula for the digammafunction, ψ[(n + 1)/2] = 2ψ(n) − ψ(n/2) − 2 ln 2. The result is

I3(n) = 2n

{[1 + (−1)n]ψ(n) − (−1)nψ

(n

2

)+ γ + [1 − (−1)n] ln 2

}−

− 2n + 1 − (−1)n, (A22)

which we find to be very useful in Appendix E.Since we have developed many estimates based upon the digamma function, it

may be useful to record another inequality for this function. By way of Binet’s firstformula [20]

ln �(z) = (z − 12) ln z − z + 1

2 ln 2π +∫ ∞

0

(1

2− 1

t+ 1

et − 1

)e−tz

tdt (A23)

228 MARK W. COFFEY

we have

ψ(z) − ln z = −1

z+

∫ ∞

0

(1

t− 1

et − 1

)e−tz dt. (A24)

By performing manipulations on this representation, it is then possible to show that

− 1

2z− 1

12z2< ψ(z) − ln z < − 1

2z. (A25)

The right inequality in (A25) also follows immediately from the fact 1/t − 1/

(et − 1) < 1/2 proved in [10].Another way to proceed in estimating a finite alternating sum such as S1 is to

rewrite it as a contour integral [18, 30],

N∑

k=�

(N

k

)(−1)kf (k) = − 1

2πi

∫

C

B(N + 1, −z)f (z) dz, (A26)

where C is a positively oriented closed curve surrounding the points �, �+1, . . . , N ,B(x, y) = �(x)�(y)/�(x +y) is the beta function, and f (z) is an analytic contin-uation of the discrete sequence f (k) to the complex plane, with no poles within theregion surrounded by C. When the integrand decreases sufficiently rapidly toward±i∞, the asymptotic evaluation of this expression can be achieved by extendingthe contour of the integral to the left and collecting the residues at the newly en-countered poles. However, in this paper we have been interested in gaining moreinformation than just an asymptotic evaluation.

Appendix B: Alternative Evaluation of the Summation (30)

Here we perform the sum of Equation (30) by using the integral representation ofEquation (38) for the zeta function. We also record two simpler sums which areuseful in the proof of Theorem 3.

Upon substituting Equation (38) into the left side of Equation (30) we find that∞∑

n=k

ζ(n)

2n

(n − 1

k − 1

)= 1

2k+1(k − 1)!∫ ∞

0

tk−1 dt

sinh(t/2). (B1)

With a change of variable and evaluation of the integral [20] we have∞∑

n=k

ζ(n)

2n

(n − 1

k − 1

)= 1

2(k − 1)!∫ ∞

0

uk−1 du

sinh u= (1 − 2−k)ζ(k). (B2)

Then∞∑

n=k

[ζ(n) − 1]2n

(n − 1

k − 1

)= (1 − 2−k)ζ(k) − 1

2k

∞∑

n=0

1

2n

(n + k − 1

n

)

= (1 − 2−k)ζ(k) − 1, (B3)


which is Equation (30).In connection with the proof of Theorem 3, and Equation (33) in particular, we

write two other zeta function sums:

∞∑

n=2

ζ(n) − 1

n2n= 1 − γ

2+ ln

√π

2, (B4a)

∞∑

n=2

ζ(n) − 1

2n= ln 2 − 1

2 . (B4b)

Appendix C: Alternative Proof of Theorem 3

We proceed to deduce Theorem 3 in a way similar to the proof of Theorem 5of [39]. In the process, we correct a typographical error which appears in both thatproof and Remark 5 and Equation (1.11) of this reference.

From the definition of the xi function in terms of the zeta function and Equa-tion (28), we have

ξ ′(s)ξ(s)

= 1

s− 1

2 ln π + 12ψ

(s

2

)+ ζ ′(s)

ζ(s)+ 1

s − 1

=∞∑

k=0

(−1)kσk+1(s − 1)k, (C1)

where the sum of the last two terms on the left side is given by Equation (12) andsimply 1/s = ∑∞

j=0(1 − s)j for |1 − s| < 1. From the expansion [20]

ψ(x) = −γ +∞∑

k=2

(−1)kζ(k)(x − 1)k−1, (C2)

and the doubling formula satisfied by the digamma function, ψ(2z) = 12 [ψ(z) +

ψ(z + 1/2)] + ln 2, we obtain

12ψ

(s

2

)= −γ

2+

∞∑

k=1

(−1)k+1ζ(k + 1)(1 − 2−k−1)(s − 1)k − ln 2. (C3)

The substitution of Equation (C3) into Equation (C1) and the equating of coeffi-cients of like powers of s − 1 gives again Equation (33) for σ1 from the constantterm and Equation (29) from the rest of the terms. This gives Theorem 3, linkingsums of reciprocal powers of the complex zeros of the zeta function with the se-quence {ηj } appearing in the expansion (12) of the logarithmic derivative of thezeta function

230 MARK W. COFFEY

Appendix D: Tabulated Numerical Values

k λk ηk k λk ηk

0 −0.577216 33 21.9582 5.99622 × 10−17

1 0.0230957 0.187546 34 23.1301 −1.99874 × 10−17

2 0.0923457 −0.0516886 35 24.3188 6.66247 × 10−18

3 0.207639 0.0147517 36 25.5232 −2.22082 × 10−18

4 0.368793 −0.00452448 37 26.7422 7.40274 × 10−19

5 0.575543 0.0014468 38 27.9749 −2.46755 × 10−19

6 0.827566 −0.000471544 39 29.2202 8.22527 × 10−20

7 1.12446 0.00015518 40 30.4774 −2.74176 × 10−20

8 1.46576 −0.0000513452 41 31.7454 9.13919 × 10−21

9 1.85092 0.0000170414 42 33.0236 −3.0464 × 10−21

10 2.27934 −5.66605 × 10−6 43 34.3111 1.01547 × 10−21

11 2.75036 1.88585 × 10−6 44 35.6072 −3.38488 × 10−22

12 3.26326 −6.28055 × 10−7 45 36.9113 1.12829 × 10−22

13 3.81724 2.09241 × 10−7 46 38.2227 −3.76098 × 10−23

14 4.41148 −6.97247 × 10−8 47 39.5408 1.25366 × 10−23

15 5.04508 2.32372 × 10−8 48 40.8653 −4.17887 × 10−24

16 5.71711 −7.74484 × 10−9 49 42.1955 1.39295 × 10−24

17 6.42658 2.58144 × 10−9 50 43.5311 −4.64318 × 10−25

18 7.17248 −8.60444 × 10−10 51 44.8718 1.54773 × 10−25

19 7.95374 2.86808 × 10−10 52 46.2172 −5.15909 × 10−26

20 8.76928 −9.56012 × 10−11 53 47.5671 1.71970 × 10−26

21 9.61796 3.18668 × 10−11 54 48.9214 −5.73232 × 10−27

22 10.4986 −1.06222 × 10−11 55 50.2798 1.91077 × 10−27

23 11.4101 3.54072 × 10−12 56 51.6423 −6.36925 × 10−28

24 12.3513 −1.18024 × 10−12 57 53.0089 2.12308 × 10−28

25 13.3210 3.93412 × 10−13 58 54.3795 −7.07695 × 10−29

26 14.3179 −1.31137 × 10−13 59 55.7542 2.35898 × 10−29

27 15.3408 4.37124 × 10−14 60 57.1331 −7.86327 × 10−30

28 16.3885 −1.45708 × 10−14 61 58.5163 2.62109 × 10−30

29 17.4599 4.85694 × 10−15 62 59.9039 −8.73697 × 10−31

30 18.5538 −1.61898 × 10−15 63 61.2962 2.91232 × 10−31

31 19.6689 5.3966 × 10−16 64 62.6934 −9.70775 × 10−32

32 20.8041 −1.79887 × 10−16 65 64.0957 3.23591 × 10−32


k λk ηk k λk ηk

66 65.5033 −1.07864 × 10−32 84 92.1386 −2.78415 × 10−41

67 66.9167 3.59546 × 10−33 85 93.7099 9.28051 × 10−42

68 68.3361 −1.19849 × 10−33 86 95.2924 −3.09350 × 10−42

69 69.7618 3.99496 × 10−34 87 96.8862 1.03117 × 10−42

70 71.1942 −1.33165 × 10−34 88 98.4912 −3.43723 × 10−43

71 72.6337 4.43884 × 10−35 89 100.1076 1.14574 × 10−43

72 74.0805 −1.47961 × 10−35 90 101.7352 −3.81914 × 10−44

73 75.5350 4.93205 × 10−36 91 103.3741 1.27304 × 10−44

74 76.9976 −1.64402 × 10−36 92 105.0242 −4.24349 × 10−45

75 78.4686 5.48005 × 10−37 93 106.6852 1.41450 × 10−45

76 79.9484 −1.82668 × 10−37 94 108.3572 −4.71500 × 10−46

77 81.4373 6.08895 × 10−38 95 110.0398 1.57166 × 10−46

78 82.9357 −2.02964 × 10−38 96 111.7328 −5.23888 × 10−47

79 84.4437 6.76550 × 10−39 97 113.4361 1.74629 × 10−47

80 85.9617 −2.25516 × 10−39 98 115.1492 −5.82098 × 10−48

81 87.4900 7.51722 × 10−40 99 116.8719 1.94032 × 10−48

82 89.0288 −2.50574 × 10−40 100 118.6038 −6.46775 × 10−49

83 90.5782 8.35246 × 10−41

Appendix EI: Sum Estimations and Lower Bounds Pertinent to OtherDirichlet Functions

Analogous to the corollary which we have presented below Equation (24) of thetext, here we develop similar lower bounds appropriate for explicit formulas forDirichlet and Hecke L-functions. We make substantial use of the very recent resultsof [34], of which we need to recall some details. We relegate to the end of this firstpart of the appendix some relations concerning elementary sums. In the secondpart, we provide independent derivations of the major results, Theorems 1 and 2,of [34].

Let χ be a primitive Dirichlet character of modulus r , and L(s, χ) the DirichletL-function of character χ . The function

ξ(s, χ) =(

π

r

)−(s+a)/2

�

(s + a

2

)L(s, χ), (E1)

where a is 0 if χ(−1) = 1 and a is 1 if χ(−1) = −1, satisfies the functionalequation ξ(s, χ) = εχξ(1 − s, χ), with εχ a constant of absolute value one. The

232 MARK W. COFFEY

function ξ(s, χ) is an entire function of order one and has a product representationξ(s, χ) = ξ(0, χ)

∏ρ(1 − s/ρ), where the product is over all the zeros of ξ(s, χ).

We put

λχ(n) =∑

ρ

[1 −

(1 − 1

ρ

)n], n � 1. (E2)

We presume that λχ(n) > 0 for all n = 1, 2, . . . if and only if all of the zeros ofξ(s, χ) are located on the critical line Re s = 1/2. Then, Li has obtained [34]

λχ(n) = Sχ(n) + n

2

(ln

r

π− γ

)+ τχ(n), (E3)

where

Sχ(n) ≡ −n∑

j=1

(n

j

)(−1)j−1

(j − 1)!∞∑

k=1

�(k)

kχ(k)(ln k)j−1,

= −∞∑

k=1

�(k)

kχ(k)L1

n−1(ln k), (E4)

τχ(n) =n∑

j=2

(n

j

)(−1)j (1 − 2−j )ζ(j) −

− n

2

∞∑

�=1

1

�(2� − 1)for χ(−1) = 1,

=n∑

j=2

(n

j

)(−1)j 2−j ζ(j) for χ(−1) = −1, (E5)

and Lαn is an associated Laguerre polynomial. We recall Equation (25) of the text,

τχ(n) = S1 − n ln 2 if χ(−1) = 1,

= S0 − S1 if χ(−1) = −1, (E6)

where S0 is defined in Equation (A15) of Appendix A. Therefore, by using thesummation estimations presented in Appendix A, we obtain

λχ � Sχ(n) + n

2ln n +

+ n

2

(ln

r

π− 1 − 2 ln 2

)+ 1

2 for χ(−1) = 1,

� Sχ(n) + n

2ln n + (E7)

+ n

2

(ln

r

π− 1

)+ 1

2 for χ(−1) = −1.


In accord with the discussion of the text, we conjecture that the sum Sχ is ‘small’.By this we mean that Sχ(n) could be O(n) and probably even Sχ is O(n1/2+ε), forε > 0. This could result from a near exponential amount of cancellation in thissum due to the phases present in the Dirichlet characters.

We next introduce the function

ξE(s) = cENs/2(2π)−s�(s + 12)LE(s + 1

2), (E8)

where LE is the L-series associated with an elliptic curve E over the rationalnumbers, N is the conductor, and cE is a constant chosen so that ξE(1) = 1[9, 37]. The function of Equation (E8) is an entire function of order one and satisfiesξE(s) = wξE(1 − s) where w = (−1)r with r being the vanishing order of ξE(s)

at s = 1/2.We let

λE(n) =∑

ρ

[1 −

(1 − 1

ρ

)n], n � 1, (E9)

where the sum is over all zeros ρ of ξE(s). All of these zeros lie on the critical lineif and only if [34] λE(n) > 0 for all n = 1, 2, . . . .

Now Li [34] has obtained the explicit formula

λE(n) = SE(n) + n

(ln

√N

2π− γ

)+ n

(−2

3+

∞∑

�=1

3

�(2� + 3)

)+

+n∑

j=2

(n

j

)(−1)j

∞∑

�=1

1

(� + 1/2)j, (E10)

where

SE(n) ≡ −n∑

j=1

(n

j

)(−1)j−1

(j − 1)!∞∑

k=1

�(k)

k3/2b(k)(ln k)j−1,

= −∞∑

k=1

�(k)

k3/2b(k)L1

n−1(ln k). (E11)

In Equation (E11), b(pk) = akp if p|N and b(pk) = αk

p + βkp if (p, N) = 1, where

for each prime number p, αp and βp are the roots of the equation T 2 − apT + p

and the values of ap are connected with the reduction of E at p [34].We recall Equation (26) of the text, so that we may write

λE(n) = SE(n) + n

(ln

√N

2π− γ

)+

+ 2(1 − 13 ln 2)n + S3 − S0 + 2n + (−1)n − 1, (E12)

234 MARK W. COFFEY

where the zeta function sum S3 is defined in Equation (A18) of Appendix A (andsee below, Equation (E15)). Then, by the results of Appendix A we obtain

λE(n) � SE(n) + n ln n + n ln

√N

2π+ (3 + 4

3 ln 2)n − 1. (E13)

Again, we conjecture that the sum SE is O(n1/2+ε). The values of ap include 0and ±1, so that the values of b(pk) can either be zero or include significant signalternation when p|N . Similarly, for (p, N) = 1, the roots of T 2 − apT + p caninclude ±√

p and [±1±√1 − 4p]/2, giving various sign changes in b(pk). When

E has good reduction at p, −2√

p � ap � 2√

p, so that it again appears that b(pk)

can have significant changes in sign, possibly leading to much cancellation in SE .Concerning Equations (E6) and (E12) we record and briefly discuss some ele-

mentary summation results. We have

ln 2 =∞∑

n=1

1

2n − 1− 1

2

∞∑

n=1

1

n, (E14a)

giving

ln 2 = 43 − 3

∞∑

n=1

1

2n(2n + 3), (E14b)

leading to

− 23 +

∞∑

n=1

3

n(2n + 3)= 2(1 − ln 2). (E14c)

In addition, we have

∞∑

�=1

1

(� + 1/2)j=

∞∑

m=1

2j

(2m + 1)j

=∞∑

m=1

(2

2m + 1

)j

+∞∑

m=1

(2

2m

)j

− ζ(j)

=∞∑

k=3,odd

(2

k

)j

+∞∑

k=2,even

(2

k

)j

− ζ(j)

=∞∑

k=2

(2

k

)j

− ζ(j) = 2j [ζ(j) − 1] − ζ(j). (E15)

This equation is a restatement of the relation between the Riemann zeta functionand the Hurwitz zeta function ζ(s, a): ζ(s) = ζ(s, 1) = (2s − 1)−1ζ(s, 1/2).


The sum of Equation (E15) also has a close relation to ψ(n)(1/2), where ψ(j) isthe polygamma function, because [20]

ψ(n)( 12) = (−1)n+1n!

∞∑

k=0

1

(k + 1/2)n+1. (E16a)

Then, with the use of Equation (E15), we have

ψ(n)( 12) = (−1)n+1n!(2n+1 − 1)ζ(n + 1), (E16b)

which is the expected result. In general, we have ψ(n)(x) = (−1)n+1n!ζ(n + 1, x).We may also write an integral representation for the polygamma function which

is very useful for evaluating terms in explicit formulas for sums over zeros of zetafunctions. By differentiating an integral representation for ψ(z) + γ , we have

ψ(m−1)(z) = (−1)m

∫ ∞

0

e−zt tm−1

1 − e−tdt

= (−1)m

2

∫ ∞

0

tm−1e−(z−1/2)t

sinh(t/2)dt, (E17)

giving the specific values

ψ(m−1)( 12) = (−1)m2m−1

∫ ∞

0

ym−1

sinh ydy, (E18a)

ψ(m−1)(1) = (−1)m2m−1∫ ∞

0

ym−1e−y

sinh ydy, (E18b)

and

ψ(m−1)( 32) = (−1)m

2

∫ ∞

0

tm−1e−t

sinh(t/2)dt. (E18c)

Appendix EII: Explicit Formulas for Dirichlet and Hecke L-Functions

Here we give alternative derivations of the very recent main results of Li [34],Theorems 1 and 2, of [34]. The procedure is very similar to the proof of Theorem 1of the text. The Riemann zeta function case extends since the Dirichlet and HeckeL-functions also have product expansions over their zeros and have explicit formsof their logarithmic derivatives. These derivations also make it very apparent thatcertain polygamma constants are the source of the elementary sums described inthe first part of this appendix.

Due to the product expansion of ξ(s, χ), we have the formula

λχ(n) =n∑

m=1

(n

m

)1

(m − 1)![

dm

dsmln ξ(s, χ)

]

s=1

. (E19)

236 MARK W. COFFEY

This equation is the analog of Equation (15) of the text or Equation (G5) of Ap-pendix G for the Riemann zeta function case. From Equation (E1) we have

ln ξ(s, χ) = −(s + a)

2ln

(π

r

)+ ln �

(s + a

2

)+ ln L(s, χ), (E20)

giving

d

dsln ξ(s, χ) = ln

(r

π

)+ 1

2ψ

(s + a

2

)−

−∞∑

n=1

�(n)χ(n)

ns, Re s > 1, (E21)

where ψ = �′/� is the digamma function and � is the von Mangoldt function,such that �(k) = ln p when k is a power of a prime and �(k) = 0 otherwise. Form � 2, we then have

[ln ξ(s, χ)](m) = 1

2mψ(m−1)

(s + a

2

)−

− (−1)m−1∞∑

n=1

�(n)χ(n)

nslnm−1 n, (E22)

where ψ(n) is again the polygamma function. By taking the limit s → 1 in Equa-tion (E22) we then obtain the representation

λχ(n) =[

ln

(r

π

)+ ψ

(a + 1

2

)]n

2+

+n∑

m=2

(n

m

)1

(m − 1)!2−mψ(m−1)

(a + 1

2

)−

−n∑

m=1

(n

m

)(−1)m−1

(m − 1)!∞∑

n=1

�(n)χ(n)

nlnm−1 n, (E23)

where ψ(1/2) = −γ − 2 ln 2, ψ(1) = −γ , γ is the Euler constant, ψ(m−1)(1) =(−1)m(m − 1)!ζ(m), and ψ(m−1)(1/2) is given in Equation (E16b). The infiniteseries in the sum Sχ(n) is convergent by the prime number theorem for arithmeticprogressions [15, 34]. We have therefore obtained the result Equation (E3).

Similarly, due to the product expansion of ξE(s), we have the formula

λE(n) =n∑

m=1

(n

m

)1

(m − 1)![

dm

dsmln ξE(s)

]

s=1

, (E24)

where from Equation (E8) we have

ln ξE(s) = ln cE + s

2ln N − s ln 2π + ln �(s + 1

2) + ln LE(s + 12), (E25)


and

d

dsln ξE(s) = 1

2 ln N − ln 2π + ψ(s + 12) −

−∞∑

n=1

�(n)b(n)

ns+1/2, Re s > 1, (E26)

where b(n) is discussed in the first part of this appendix. For m � 2, we then have

[ln ξE(s)](m) = ψ(m−1)(s + 12) − (−1)m−1

∞∑

n=1

�(n)b(n)

ns+1/2lnm−1 n. (E27)

By taking the limit s → 1 in Equation (E27) we then obtain the representation

λE(n) =[

ln

(√N

2π

)+ ψ( 3

2)

]n +

n∑

m=2

(n

m

)1

(m − 1)!ψ(m−1)( 3

2) −

−n∑

m=1

(n

m

)(−1)m−1

(m − 1)!∞∑

n=1

�(n)b(n)

n3/2lnm−1 n, (E28)

where ψ(3/2) = 2(1 − ln 2) − γ and ψ(m−1)(3/2) = (−1)m(m − 1)![2m(ζ(m) −1) − ζ(m)], giving the result Equation (E12).

Finally, we consider the case of the Dedekind zeta function ζk, for which weneed to introduce some additional notation. We let k be an algebraic number fieldwith r1 real places, r2 imaginary places, and degree n = r1 +2r2. The zeta functionζk has the product expansion ζk(s) = ∏

p(1 − Np−s)−1 for Re s > 1, where theproduct is taken over all finite prime divisors of k. We put G1(s) = π−s/2�(s/2)

and G2(s) = (2π)1−s�(s), so that obviously G1(1) = G2(1) = 1. Then thefunction

Zk(s) ≡ Gr11 (s)G

r22 (s)ζk(s) (E29)

satisfies the functional equation Zk(s) = |dk|1/2−sZk(1 − s), where dk is thediscriminant of k.

We let ck = 2r1(2π)r2hR/e, where h, R, and e are respectively the number ofideal classes of k, the regulator of k, and the number of roots of unity in k. With

ξk(s) ≡ c−1k s(s − 1)|dk|s/2Zk(s), (E30)

this function is entire and has ξk(0) = 1 [40, 48]. We first present a motivationthat an explicit formula analogous to that for λE and λχ exists, and then elsewheredevelop the corresponding explicit formula, putting

λn =n∑

m=1

(n

m

)1

(m − 1)![

dm

dsmln ξk(s)

]

s=1

. (E31)

238 MARK W. COFFEY

From Equations (E29) and (E30) we have

ln ξk(s) = − ln ck + ln s + ln(s − 1) + s

2ln |dk| +

+ r1

[− s

2ln π − ln �

(s

2

)]+

+ r2[(1 − s) ln(2π) + ln �(s)] + ln ζk(s), (E32)

and

d

dsln ξk(s) = 1

s+ 1

s − 1+ 1

2 ln |dk| + r1

2

[− ln π + ψ

(s

2

)]+

+ r2[− ln(2π) + ψ(s)] + ζ ′k(s)

ζk(s), Re s > 1, (E33)

where [42]

ζ ′k(s)

ζk(s)= −

∑

p

∞∑

m=1

ln Np

Npms, Re s > 1. (E34)

In Equation (E34), p runs over the prime ideals of k and N represents the norm.For m � 2, we then have

dm

dsmln ξk(s) = (−1)m(m − 1)!

sm−1+ (−1)m(m − 1)!

(s − 1)m−1+ r1

2mψ(m−1)

(s

2

)+

+ r2ψ(m−1)(s) + dm−1

dsm−1

[ζ ′k(s)

ζk(s)

], Re s > 1, (E35)

where the evaluation of the first, third and fourth terms on the right side of Equa-tion (E35) at s = 1 gives the contribution to λn of

λ(ψ)n =

n∑

m=2

(−1)m

(n

m

){1 + [(1 − 2−m)r1 + r2]ζ(m)}. (E36)

The evaluation of all of the terms on the right side of Equation (E33) but the secondand last at s = 1 gives to λn the contribution n[1 + 1

2 ln |dk| − n2 (ln π + γ ) −

(r1 + r2) ln 2]. With the aid of Equation (E34) we have

dm−1

dsm−1

[ζ ′k(s)

ζk(s)

]= −

∑

p

∞∑

�=1

(−1)m−1�m−1 lnm Np

Np�s, Re s > 1. (E37)

Taking the limit s → 1 in Equations (E33)–(E35) and (E37) should yield thefinal explicit representation for λn, subject to justification of the convergence ofthe resulting series.


Appendix F: Further Riemann Zeta Function Sum Estimations

As a generalization of sums such as S1 of Equation (16) of the text and S0 ofEquation (A15) of Appendix A, we consider here sums of the form

Sν(κ, n) ≡n∑

m=ν+2

(−1)m

(n

m

)κmζ(m − ν), (F1)

where ν + 2 � n, which can be extended to |Re ν| + 2 � n. In Equation (F1)we have introduced both the positive multiplier κ and shift ν. The special cases ofEquation (F1) of direct interest to this paper are ν = 0 with κ = 1 or κ = 2±1. Weare interested to both reformulate the sum Sν and to obtain a lower bound for it.

If we reorder the two sums in Equation (F1), the inner sum takes the form

n∑

m=ν+2

(−1)m

(n

m

)κm

jm−ν

= (−1)ν κν+2n!j 2

n−ν−2∑

m=0

(1)m

(n − m − ν − 2)!(m + ν + 2)!1

m!(

κ

j

)m

. (F2)

If we use the relations (m + ν + 2)! = (ν + 2)!(ν + 3)m and (n − m − ν − 2)! =(n− ν − 2)!/(2 + ν −n)m, where the Pochhammer symbol (z)n = �(z+n)/�(z),we obtain the terminating hypergeometric form

n∑

m=ν+2

(−1)m

(n

m

)κm

jm−ν

= (−1)ν

(n

ν + 2

)κν+2

j 2 2F1

(1, 2 − n + ν; ν + 3; κ

j

), (F3)

giving

Sν(κ, n) = (−1)ν

(n

ν + 2

)κν+2

∞∑

j=1

1

j 2 2F1

(1, 2 − n + ν; ν + 3; κ

j

). (F4)

We now define the comparison integral

Iν(κ, n)

≡ (−1)ν

(n

ν + 2

)κν+2

∫ ∞

1

1

j 2 2F1

(1, 2 − n + ν; ν + 3; κ

j

)dj. (F5)

With the change of variable v = 1/j the integration is easily accomplished [20] interms of the generalized hypergeometric function pFq [4]:

Iν(κ, n) = (−1)ν

(n

ν + 2

)κν+2

3F2(1, 1, ν + 2 − n; 2, ν + 3; κ). (F6)

240 MARK W. COFFEY

Equation (F6) is simply the result of term-by-term integration and the fact that(1)k/(2)k = 1/(k + 1). The relation Sν(κ, n) � Iν(κ, n) then yields a family ofinequalities.

When κ is unity we have the reduction ψ(z) = (z−1) 3F2(1, 1, 2−z; 2, 2; 1)−γ and therefore

Iν(1, n) = (−1)ν

(n

ν + 1

)[ψ(n + 1) + γ − Hν+1], (F7)

where ψ is the digamma function and Hn = ∑nk=1 1/k is the nth harmonic

number [11], which can also be written as

Iν(1, n) = (−1)ν

(n

ν + 1

)[Hn − Hν+1]. (F8)

For ν = 0, this gives the result of Equation (A17). An alternative form of Iν(1, n)

can be obtained by applying Theorem 1 of [44]:

Iν(1, n) = (−1)ν

(n

ν + 1

)[ψ(n − ν) − ψ(ν + 2) −

ν+1∑

k=1

(−ν − 1)k

k(n − ν)k

], (F9)

valid for n > 0 and integral ν � −1. The relations above at unit argument can alsobe looked upon as special cases of [35]

3F2(1, 1, ν + 1; 2, λ + 1; 1)

= λ

ν[ψ(λ) − ψ(λ − ν)], ν �= 0, Re (λ − ν) > 0,

(F10)= λψ ′(λ), ν = 0, Re (λ) > 0,

i.e.,

∞∑

n=1

(ν)n

n(λ)n

= ψ(λ) − ψ(λ − ν), Re (λ − ν) > 0, λ �= 0, −1, −2, . . . . (F11)

For κ = 1/2 we obtain

Iν(1/2, n) = (−1)ν

(n

ν + 2

)1

2ν+2 3F2(1, 1, ν + 2 − n; 2, ν + 3; 12), (F12)

which for ν = 0 is in agreement with I0 − I1, where I1 is given in Equation (24) ofthe text. For κ = 2 we obtain

Iν(2, n) = (−1)ν

(n

ν + 2

)2ν+2

3F2(1, 1, ν + 2 − n; 2, ν + 3; 2), (F13)

which is in agreement with I3 of Equation (A22) of Appendix A when ν = 0.


5. Appendix G: Alternative Representation of Li’s λj ’s

Here we extend Theorem 1, based upon an expansion of the logarithmic derivativeof the Riemann zeta function with a larger radius of convergence than Equation (12)of the text. We demonstrate the following representation,

λn = −n∑

m=1

(n

m

)η

(12)

m−1 + 7 −

− [( 2

3)n + ( 4

5)n + ( 6

7)n + ( 8

9)n + ( 10

11)n + ( 12

13)n] +

+n∑

m=2

(−1)m

(n

m

)(1 − 2−m)ζ(m) − n

2(γ + ln π + 2 ln 2), (G1)

where we currently do not have an arithmetic interpretation of the constants η(12)j .

From the expansion around s = 1 of the logarithmic derivative of the zetafunction,

ζ ′(s)ζ(s)

= − 1

(s − 1)+

6∑

j=1

1

(s + 2j)−

∞∑

p=0

η(12)p (s − 1)p, (G2)

we have

ln ζ(s) = − ln(s − 1) +6∑

j=1

ln(s + 2j) −

−∞∑

p=1

η(12)p−1

p(s − 1)p + constant, (G3)

giving

ln ξ(s) = − ln 2 + ln s − s

2ln π + ln �

(s

2

)+ constant +

+6∑

j=1

ln(s + 2j) −∞∑

p=1

η(12)

p−1

p(s − 1)p. (G4)

With the expansion (G2) the radius of convergence has been increased to 13 (seeFigure 1), as we have included the contribution of all trivial zeros of ζ prior to theencounter with the first complex zero ρ1. We next evaluate

λn =n∑

m=1

(n

m

)1

(m − 1)![

dm

dsmln ξ(s)

]

s=1

, (G5)

using again the special values ψ(1/2) = −γ − 2 ln 2 and ψ(n)(1/2) =(−1)n+1n!(2n+1 −1)ζ(n+1) for n � 1, where ψ = �′/� is the digamma functionand ψ(j) is the polygamma function. Recalling the relations

dj

dsjln(s + 2k) = −(−1)j (j − 1)!

(s + 2k)j, j � 1, (G6)

242 MARK W. COFFEY

Figure 1. Diagram of consecutively increasing circles of convergence corresponding to theinclusion of the first six trivial zeros of the Riemann zeta function in the expansion (G2). Thelocation of the first nontrivial zero ρ1 is indicated.

and (dj /dsj )(s − 1)k = 0 for k < j , and the sumn∑

m=1

(n

m

)(−1

k

)m

= −1 +(

k − 1

k

)n

, (G7)

we find Equation (G1). The constant term of 6 = 7−1 in Equation (G1) serves as acount of the number of trivial zeros of ζ accounted for in the expansion (G2) whilethe additional explicit negative terms beyond Equation (10) of the text appearingthere are exponentially decreasing with n. We have developed in Equation (G2) anexpansion with coefficients η

(12)j whose magnitudes increase no faster than 1/13j

for large j .By using Equation (20) for S1, an extension of the corollary of the text is

COROLLARY G1.

λn � n

2ln n − (1 + ln π + 2 ln 2)

n

2+ 15

2 −


− [( 2

3)n + ( 4

5)n + ( 6

7)n + ( 8

9)n + ( 10

11)n + ( 12

13)n] − |S(12)

2 |, (G8)

where we have put S(12)

2 ≡ − ∑nm=1

(n

m

)η

(12)

m−1.

In developing expansions such as

ζ(s → −2k) = ζ ′(−2k)(s + 2k) + 12ζ

′′(−2k)(s + 2k)2 ++ 1

6ζ′′′(−2k)(s + 2k)3 + O[(s + 2k)4], k � 1, (G9)

and

ζ ′(s)ζ(s)

= 1

(s + 2k)+ ζ ′′(−2k)

2ζ ′(−2k)+

+[− [ζ ′′(−2k)]2

4[ζ ′(−2k)]2+ ζ ′′′(−2k)

3ζ ′(−2k)

](s + 2k) + O[(s + 2k)2], (G10)

it is useful to have the derivatives

ζ ′(−2n) = (−1)n (2n)!2(2π)2n

ζ(2n + 1), n � 1, (G11)

and

ζ ′′(−2n) = (−1)n (2n)!2(2π)2n

{[ln(4π2) − 2ψ(2n) − 1

n

]ζ(2n + 1) −

− 2ζ ′(2n + 1)

}, n � 1, (G12)

which follow easily by differentiating the functional equation for ζ(s) and puttings = 2n + 1.

We note in passing the numerical value of the constant

1

2

6∑

k=1

ζ ′′(−2k)

ζ ′(−2k)= − 81959

5544 + 6(γ + ln 2π) −

−6∑

k=1

ζ ′(2k + 1)

ζ(2k + 1)� −0.0926073. (G13)

We see from Equation (G2) that the value of η(12)

0 is given by

η(12)

0 = γ −6∑

k=1

1

2k + 1� 0.377918. (G14)

Therefore we can modify Equation (G8) to

244 MARK W. COFFEY

COROLLARY G2.

λn � n

2ln n − (1 + ln π + 2 ln 2)

n

2+ 15

2 −− [

( 23)

n + ( 45)

n + ( 67)

n + ( 89)

n + ( 1011)

n + ( 1213)

n] − |η(12)

0 |n −

−∣∣∣∣∣

n∑

m=2

(n

m

)η

(12)

m−1

∣∣∣∣∣. (G15)

Appendix H: On Derivatives of the Riemann Zeta Function

Here we capture various formulas for integer order derivatives of the zeta function.We anticipate that these could be useful in further development of the discretemoment problem for the coefficients ηj of Equations (10)–(14) and Equation (41)and η

(12)j of Equations (G1)–(G4) of Appendix G.

We first note that the functional equation for ζ , along with the evaluationζ(1 − 2n) = −B2n/2n for n � 1, where Bn are Bernoulli numbers, yields

ζ ′(−1) = ζ ′(2)

2π2+ 1

12(ln 2π + γ ), (H1)

and

ζ ′′(−1) = 1

6

(π2

8− ln2 2

2

)+

[(1 − γ )

π2

6+ ζ ′(2)

]ln 2

π2−

− 1 − γ

π2ζ ′(2) − 1

2π2ζ ′′(2) −

− 1

12

[−1 + (1 − γ )2 + π2

6

]+ 1

12 ln2 π + 2 ln πζ ′(−1), (H2)

where γ is the Euler constant, and this can be continued to higher order derivatives.The derived functional equation upon which Equations (H1) and (H2) are based is

21−s�(s)ζ(s) cos

(π

2s

)[ψ(s) − ln 2] + 21−s�(s)ζ ′(s) cos

(π

2s

)−

− π

221−s�(s)ζ(s) sin

(π

2s

)= πs ln πζ(1 − s) − πsζ ′(1 − s). (H3)

Now Elizalde [17] has given an expression for ζ ′(−m, q), where ζ(z, q) is theHurwitz zeta function, valid for any negative integer value of z. We state herespecial cases for q = 1:

ζ ′(−1) = − 16 − 1

2

∞∑

k=1

B2k+2

k(2k + 1)(2k + 2), (H4)

ζ ′(−2) = − 136 −

∞∑

k=1

B2k+2

(2k − 1)k(2k + 1)(2k + 2), (H5)


and

ζ ′(−k) = − 1

(k + 1)2−

−∞∑

�=1

(−1)�(2� − 1)!22�−1π2�

[min(2�−2,k)∑

r=0

(k

r

)(−1)r

(2� − r − 1)

]ζ(2�).(H6)

This means that we also have explicit expressions for ζ ′(1 + k). In particularEquations (H1) and (H4) or (H6) yield an explicit form for ζ ′(2).

On the other hand, we can employ the integral representation, Equation (38) ofthe text, to at least partially yield explicit values of the zeta derivatives for Re s > 1.Other integral representation could be used for Re s > 0, but Equation (38) servesfor illustration. In the following, ψ denotes the digamma function and ψ(j) thepolygamma function, as usual.

We have

ζ ′(s) = −ψ(s)ζ(s) + 1

�(s)

∫ ∞

0

t s−1 ln t

et − 1dt, (H7)

ζ ′′(s) = −ψ ′(s)ζ(s) + ψ2(s)ζ(s) − 2ψ(s)

�(s)

∫ ∞

0

t s−1 ln t

et − 1dt +

+ 1

�(s)

∫ ∞

0

t s−1 ln2 t

et − 1dt, (H8)

and therefore

ζ ′(s)ζ(s)

= −ψ(s) + 1

�(s)ζ(s)

∫ ∞

0

t s−1 ln t

et − 1dt, (H9)

and[ζ ′(s)ζ(s)

]′= −ψ ′(s) + 1

�(s)ζ(s)

{∫ ∞

0

t s−1 ln2 t

et − 1dt −

− 1

�(s)ζ(s)

[∫ ∞

0

t s−1 ln t

et − 1dt

]2}. (H10)

This process can be continued,[ζ ′(s)ζ(s)

]′′= −ψ ′′(s) + ψ3(s) − ψ(s)ψ ′(s) − ψ ′(s)

ζ ′(s)ζ(s)

+ ψ2(s)ζ ′(s)ζ(s)

+

+ 1

�(s)ζ(s)[ψ ′(s) − ψ2(s)]

∫ ∞

0

t s−1 ln t

et − 1dt −

− 3

�2(s)ζ 2(s)

∫ ∞

0

t s−1 ln t

et − 1dt

∫ ∞

0

t s−1 ln2 t

et − 1dt +

+ 1

�(s)ζ(s)

∫ ∞

0

t s−1 ln3 t

et − 1dt +

246 MARK W. COFFEY

+ 2

�3(s)ζ 3(s)

[∫ ∞

0

t s−1 ln t

et − 1dt

]3

, (H11)

and we note that

−ψ(j)(2) = (−1)j j ![ζ(j + 1) − 1]. (H12)

Following on Equation (H7) we have

ζ (j+1)(s) = −j∑

m=0

(j

m

)ψ(j−m)(s)ζ (m)(s) +

+j∑

m=0

(j

m

)[(d

ds

)m 1

�(s)

] ∫ ∞

0

t s−1 lnj−m+1 t

et − 1dt. (H13)

The relations of this appendix can be developed much more for application toEquation (41) of the text or elsewhere.

Appendix I: A Digamma Function Integral and a Mellin Transform

Here we consider the integral

I (λ) =∫ ∞

1

(1 − x−λ)

x(x2 − 1)dx (I1)

of Equation (3.4) of [7] and evaluate it in two different ways from tabulatedresults [20]. In [7] the asymptotic behaviour of this integral for large λ was ofinterest for determining a certain limit denoted by the PF operation [Equation (3.2)there]. We also note a polynomial of [7] that can be written as a terminatingconfluent hypergeometric series. This associated Laguerre polynomial was usedin calculating forward and inverse Mellin transforms and we give alternative trans-forms.

With the change of variable v(x) = x−1 in Equation (I1) we have

I (λ) =∫ ∞

0

v(1 − vλ)

(1 − v2)dv. (I2)

From [20] we then obtain

I (λ) = 1

2

[ψ

(λ

2+ 1

)+ γ

]= 1

2

[ψ

(λ

2

)+ γ + 2

λ

], λ > −2, (I3)

where ψ = �′/� is the digamma function. On the other hand, we may employ apartial fractional decomposition in Equation (I2) and another tabulated result [20]so that

I (λ) = 1

2

∫ 1

0(1 − vλ)

[1

(1 − v)− 1

(1 + v)

]dv

= 1

2

[ψ(λ + 1) + γ − ln 2 +

∫ 1

0

vλ

1 + vdv

], (I4)


where the last integral is given by [20]

β(λ + 1) = 1

2

[ψ

(λ

2+ 1

)− ψ

(λ + 1

2

)]. (I5)

Use of the doubling formula ψ[(λ + 1)/2] = 2ψ(λ) − ψ(λ/2) − 2 ln 2 then againyields Equation (I3). In Appendix A we have additionally given many inequalitiesfor the digamma function.

The polynomial

Pn(x) =n∑

j=1

(n

j

)xj−1

(j − 1)! (I6)

was used in [7] in connection with computing Mellin transforms. By using therelations

(n

j

)= (−1)j (−n)j

j ! and (−n)j+1 = −n(1 − n)j , (I7)

where (.)n is the Pochhammer symbol, this polynomial can be written as a termi-nating confluent hypergeometric series: Pn(x) = n 1F1(1−n; 2;−x). In particular,a certain Mellin transform involving Pn converts to a Laplace transform:

∫ 1

0Pn(ln x)xs−1 dx =

∫ ∞

0Pn(−u)e−su du

= n

∫ ∞

01F1(1 − n; 2; u)e−su du

= n

sF

(1 − n, 1; 2; 1

s

)= 1 −

(1 − 1

s

)n

, (I8)

where F is the Gauss hypergeometric function [20], which is the expected re-sult [7]. In obtaining Equation (I8) we have used the reduction [20]

F

(1 − n, 1; 2;−z

t

)= (t + z)n − tn

nztn−1. (I9)

Another useful point of view of the particular polynomial (I6) is afforded bythe theory of Laguerre polynomials Ln. This family is orthogonal on the interval[0, ∞) with decaying exponential weight function. We have the relations

Pn(−x) = −dLn(x)

dx= L1

n−1(x), (I10)

where Lαn is an associated Laguerre polynomial. In addition, the recursion relations

satisfied by the Laguerre polynomials [20] give

Pn(−x) = −n

x[Ln(x) − Ln−1(x)]

= −(n + 1)

xLn+1(x) + (n + 1 − x)

xLn(x). (I11)

248 MARK W. COFFEY

Then one can recast the important Mellin transform-inverse transform pair of Equa-tion (I8):

∫ ∞

0Pn(−u)e−su du = −

∫ ∞

0

dLn(u)

due−su du

= 1 −(

1 − 1

s

)n

, Re s > 0. (I12)

In obtaining this equation, one can use integration by parts, the Laplace transformof Ln [20], and the property Ln(0) = 1. All these relations are consistent withthe connection L1

n−1(x) = n 1F1(1 − n; 2; x) and the derivative property of theconfluent hypergeometric function.

Appendix J: Regarding Conjectures 1–3

We present here plausibility arguments in possible support of our conjectures con-cerning the detailed behaviour of the sequences {σk} and {λj }. For this discussion,we let N(T ) be the number of zeros of the Riemann zeta function in the criticalstrip in the upper half plane to height T . That is, N(T ) denotes the number ofcomplex zeros in the rectangle 0 � Re s � 1 and 0 � Im s � T .

Backlund [3] showed that N(T ) satisfies

N(T ) = T

2πln

(T

2π

)− T

2π+ 7

8 + e(T ), (J1)

where

|e(T )| < 0.137 ln T + 0.443 ln ln T + 4.35 for T � 2. (J2)

We believe then that if one were to assume certain statistical properties of thedistribution of the Riemann zeros, Conjectures 2 and 3 would follow.

From Equations (J1) and (J2) we can show that there is a constant T0 such that

π [N(T + 1) − N(T )] � π ln T for T � T0. (J3)

If we write N(T ) = M(T ) + e(T ), then

M(T + 1) − M(T ) = 1

2πln

(T

2π

)+ 1

2π

(1

2T− 1

6T 2+ 1

12T 3− · · ·

)

<1

2πln

(T

2π

)+ 1

4πT. (J4)

Then with Equation (J2) we have

N(T + 1) − N(T ) <1

2πln

(T

2π

)+

+ 1

4πT0.137

[2 ln T + ln

(1 + 1

T

)]+

+ 0.866 ln ln(T + 1) + 8.70, T � 2. (J5)


Since ln(1 + 1/T ) < 1/T , we find

N(T + 1) − N(T )

< 0.433 ln T + 0.866 ln ln(T + 1) + 8.407 + 0.216

T. (J6)

We also have that

0.866 ln ln(T + 1) � 0.135 ln T for T � T0, (J7)

8.407 � 0.431 ln T for T � T0, (J8)

and we may take T0 = 3 × 108. These relations yield inequality (J3).We now assume that the Riemann hypothesis holds and consider one possible

bound that may result for the sums σk = ∑j ρ−k

j , where {ρj } represents thenontrivial zeros of the zeta function. The nontrivial zeros have the form ρj =1/2 + ε + iαj , where bounds for ε exist in the literature due to results on zero-free regions. As mentioned in the text, a zero ρj enters the sum σk along with itscomplex conjugate. We then consider the sums

∞∑

j=m

1

αkj

� 1

(k − 1)2

1 − ln([αm] − 1) + k ln([αm] − 1)

([αm] − 1)k−1, [αm] � T0, (J9)

as an approximation to σk, where [x] denotes the greatest integer contained within x.We have

∞∑

j=m

1

αkj

�∞∑

j=[αm]

∑

j�α�<j+1

1

αk�

�∞∑

j=[αm]

N(j + 1) − N(j)

jk�

∞∑

j=[αm]

ln j

jk, for [αm] � T0, (J10)

where we applied inequality (J3). Since the last sum in inequality (J10) is boundedby

∫ ∞

[αm]−1

ln u

ukdu = 1

(k − 1)2

1 − ln([αm] − 1) + k ln([αm] − 1)

([αm] − 1)k−1, (J11)

we obtain inequality (J9).

Appendix K: Observations Concerning a Dedekind Xi Function

We note here some relations concerning a Dedekind xi function ξk, extending someof our earlier derivative results for the Riemann xi function [12].

We let k be an imaginary quadratic field of discriminant d, and

�(t) = π√|d|∑

F

∞∑

m,n=−∞F(m, n)

[πt√|d|F(m, n) − 1

]×

× exp

[− 2πt√|d|F(m, n)

], (K1)

250 MARK W. COFFEY

where the first sum is over the inequivalent classes of positive definite integralquadratic forms of discriminant d. Then Li [33] has very recently shown that

ξk(s) = 4

w

∫ ∞

1�(t)(ts + t1−s) dt, (K2)

for all complex s, where w is the number of roots of unity. The function ξk is entireand satisfies ξk(s) = ξk(1 − s) and ξk(0) = ξk(1) = 2r1hR/w, where the numberof real places r1 = 0, the number of complex places r2 = 1, R = 1 is the regulator,and h is the number of ideal classes of k.

The inequivalent classes of positive definite integral quadratic forms of discrim-inant d consist of the classes represented by the forms [23, 33] F(x, y) satisfyingb2 − 4ac = d for either −a < b � a < c or 0 � b � a = c. Now it hasalso been shown [33] that F(x, y) = ax2 + bxy + cy2 satisfying this condition issuch that F(m, n) �

√|d|/2 for all integers m, n with n �= 0. It follows triviallythat πtF (m, n)/

√|d| − 1 � πt/2 − 1 > 0 for all t ∈ [1, ∞). We then have theimmediate

PROPOSITION. For all the inequivalent classes of positive definite integralquadratic forms of discriminant d, we have ξk(s) > 0 for all real s. Furthermore,the integer-order derivatives

ξ(m)k (s) = 4

w

∫ ∞

1�(t)[t s + (−1)mt1−s] lnm t dt (K3)

satisfy ξ(m)k (s) � 0 for all s � 1/2. The even order derivatives obey the condition

ξ(2m)k (s) > 0 for all s � 1/2. Of course, as also seen by the functional equation

for ξk(s), the odd-order derivatives ξ(2m+1)k vanish at s = 1/2; we have ξ

(m)k (s) =

(−1)mξ(m)k (1 − s).

This proposition seems to mean that ξk(s) has no zeros for real values of s, andextends some of the results of [12]. In turn, we may apply some of the explicitintegration results obtained there in order to evaluate the function

γ (a) ≡∫ ∞

1ω(t)

dt√t

+∫ √|d|/2a

1ω(t)

dt

t, (K4)

introduced in [33], where ω is the θ series given by ω(t) = ∑∞n=1 exp(−πn2t). We

have for the first term on the right side of Equation (K4)

∞∑

n=1

∫ ∞

1e−πn2t t−1/2 dt = 1√

π

∞∑

n=1

1

n�(1/2, n2π)

=∞∑

n=1

1

n[1 − Erf (n

√π)], (K5)


where �(x, y) is the incomplete Gamma function [20] and �(1/2, n2π) = √π [1−

2n 1F1(1/2; 3/2;−πn2)] and 1F1 is the confluent hypergeometric function [4, 20],such that 1F1(1/2; 3/2; x) = (

√π/2)Erf (

√−x)/√−x, where Erf is the error

function (probability integral) [20].For the second term on the right side of Equation (K4) we have

∞∑

n=1

∫ √|d|/2a

1e−πn2t t−1 dt =

∞∑

n=1

[Ei

(−πn2√|d|

2a

)− Ei(−πn2)

], (K6)

where Ei is the exponential integral [20]. We also have various elementary rela-tions, including

∫ √|d|/2a

1e−πn2t t−1 dt = πn2

∫ √|d|/2a

1e−πn2t ln t dt +

+ exp

(−πn2

√|d|2a

)ln

(√|d|2a

), (K7)

obtained by integration by parts, and

∫ √|d|/2a

1e−πn2t ln t dt

= 1

πn2

∫ πn2√|d|/2a

πn2e−u[ln u − ln(πn2)] dt

= 1

πn2

[∫ πn2√|d|/2a

πn2e−u ln u du−

−(

exp

(−πn2

√|d|2a

)− exp(−πn2)

)ln(πn2)

]. (K8)

Moreover, the particular derivative values

ξ(2m)k ( 1

2) = 8

w

∫ ∞

1�(t)t1/2 ln2m t dt, (K9)

and

ξ(m)k (1) = 4

w

∫ ∞

1�(t)[t + (−1)m] lnm t dt, (K10)

can be evaluated in terms of infinite series with the analytic methods of [12], andwe note that ξ

(m)k (1) > 0 for all nonnegative integers m. These special values

of Equation (K10) enter the particular logarithmic derivatives of Equation (E31)for λn.

252 MARK W. COFFEY

Appendix L: Euler–Maclaurin Summation Applied to S1(n)

Here we apply Euler–Maclaurin summation to the form of the sum S1(n) given inEquation (17) of the text. Accordingly, we define the summand function

f (k) ≡ n

2k + 1− 1 + 2nkn

(2k + 1)n, k � 0, n � 2, (L1)

such that f (0) = n − 1 and f (∞) = 0. We can write the arbitrary integer orderderivative of each term of Equation (L1). For the first term on the right side wehave

(d

dk

)jn

(2k + 1)= (−1)j 2j j !

(2k + 1)j+1n, j � 1. (L2)

When evaluated at k = 0, this term gives (−1)j 2j j !n. In regard to the last term ofEquation (L1) we have

(d

dk

)j 1

(2k + 1)n= (−1)j 2j (n)j

(2k + 1)n+j, j � 1, (L3a)

where (.)j is the Pochhammer symbol, and(

d

dk

)r

kn = n!(n − r)!k

n−r , (L3b)

which can be equally expressed as(

d

dk

)r

kn = r!(

n

r

)kn−r = (−1)r(−n)rk

n−r . (L3c)

Therefore we can write

2n

(d

dk

)�

kn(2k + 1)−n

= 2n

�∑

m=n

(�

m

)(−1)�−m2�−m(n)�−m

(2k + 1)n+�−m

n!(n − m)!k

n−m. (L4)

A version of the Euler–Maclaurin formula, given that all derivatives of f vanishat infinity, is

∞∑

n=M

f (n) =∫ ∞

M

f (x) dx −∞∑

m=1

Bm

m! f(m−1)(M), (L5a)

where Bm are Bernoulli numbers, or

∞∑

n=0

f (n) =∫ ∞

0f (x) dx + 1

2f (0) −∞∑

m=2,even

Bm

m! f(m−1)(0). (L5b)


By using Equation (24) of the text for the integral in Equation (L5b) we thereforeobtain

S1(n) = n

2[ψ(n) + γ ] −

∞∑

m=2,even

Bm

m! f(m−1)(0), (L6)

where ψ is the digamma function and γ is the Euler constant. The sums in Equa-tions (L5)–(L6) are meant in an asymptotic sense; they are highly unlikely to beconvergent. From Equation (L6) we may obtain the successive approximations

S1(n) = n

2[ψ(n) + γ ] + n

6−

∞∑

m=4,even

Bm

m! f(m−1)(0), n > 1, (L7a)

and

S1(n) = n

2[ψ(n) + γ ] + n

10−

∞∑

m=6,even

Bm

m! f(m−1)(0), n > 3. (L7b)

Equation (L7a) is expected to be a useful approximate upper bound to S1 andEquation (L7b) an approximate lower bound for this sum. When n is sufficientlylarge, only the first term on the right side of Equation (L1) contributes in Equa-tions (L6)–(L7), giving Bmf (m−1)(0)/m! = −2m−1Bm(n/m), as m − 1 is alwaysan odd integer. Given alternation in sign from B2m to B2m+2, these successive termswill change sign also. However, we emphasize that the sums in Equation (L6)–(L7)are generally divergent.

References

1. Abramowitz, M. and Stegun, I. A.: Handbook of Mathematical Functions, National Bureau ofStandards, Washington, 1964.

2. Andrews, G. E., Askey, R. and Roy, R.: Special Functions, Cambridge University Press,Cambridge, 1999.

3. Backlund, R. J.: Über die Nullstellen der Riemannschen Zetafunktion, Acta Math. 41 (1918),345–375.

4. Bailey, W. N.: Generalized Hypergeometric Series, Cambridge University Press, Cambridge,1935.

5. Berndt, B. C.: On the Hurwitz zeta function, Rocky Mountain J. Math. 2 (1972), 151–157.6. Biane, P., Pitman, J. and Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta

functions and Brownian excursions, Bull. Amer. Math. Soc. 38 (2001), 435–465.7. Bombieri, E. and Lagarias, J. C.: Complements to Li’s criterion for the Riemann hypothesis,

J. Number Theory 77 (1999), 274–287.8. Briggs, W. E.: Some constants associated with the Riemann zeta-function, Michigan Math. J.

3 (1955), 117– 121.9. Bump, D.: Automorphic Forms and Representations, Cambridge University Press, Cambridge,

1997.10. Clark, W. E. and Ismail, M. E. H.: Inequalities involving gamma and psi functions, Anal. Appl.

1 (2003), 129–140, http://www.math.usf.edu/~eclark

254 MARK W. COFFEY

11. Coffey, M. W.: On some log-cosine integrals related to ζ(2), ζ(3), and ζ(6), J. Comput. Appl.Math. 159 (2003), 205–215.

12. Coffey, M. W.: Relations and positivity results for derivatives of the Riemann ξ function, J.Comput. Appl. Math. 166 (2004), 525–534.

13. Coffey, M. W.: New results concerning power series expansions of the Riemann xi function andthe Li/Keiper constants, Preprint, 2004.

14. Coffey, M. W.: Polygamma theory, the Li/Keiper constants, and validity of the Riemannhypothesis, Preprint, 2005.

15. Davenport, H.: Multiplicative Number Theory, Springer, New York, 2000.16. Edwards, H. M.: Riemann’s Zeta Function, Academic Press, New York, 1974.17. Elizalde, E.: An asymptotic expansion for the first derivative of the generalized Riemann zeta

function, Math. Comp. 47 (1986), 347–350.18. Flajolet, P. and Sedgewick, R.: Mellin transforms and asymptotics: Finite differences and Rice’s

integrals, Theoret. Comput. Sci. 144 (1995), 101–124.19. Gourdon, X.: The 1013 first zeros of the Riemann zeta function and zeros computation at very

large height, Preprint, 2004.20. Gradshteyn, I. S. and Ryzhik, I. M.: Tables of Integrals, Series, and Products, Academic Press,

New York, 1980.21. Graham, R. L., Knuth, D. E. and Patashnik, O.: Concrete Mathematics, 2nd edn, Addison-

Wesley, Englewood Cliffs, 1994.22. Hardy, G. H.: Note on Dr. Vacca’s series for γ , Quart. J. Pure Appl. Math. 43 (1912), 215–216.23. Hua, L. K.: Introduction to Number Theory, Springer, 1982.24. Israilov, M. I.: On the Laurent Decomposition of Riemann’s zeta function, Dokl. Akad. Nauk

SSSR (Russian) 12 (1979), 9.25. Israilov, M. I.: Trudy Mat. Inst. Steklova 158 (1981), 98–104.26. Ivic, A.: The Riemann Zeta-Function, Wiley, 1985.27. Ivic, A.: The Laurent coefficients of certain Dirichlet series, Publ. Inst. Math. (Beograd) 53

(1993), 23–36. In Equation (1.7), (log x)x should be replaced with (log x)k and on p. 25“comparing with (1.15)” should be replaced with “comparing with (1.5)”.

28. Karatsuba, A. A. and Voronin, S. M.: The Riemann Zeta-Function, Walter de Gruyter, NewYork, 1992.

29. Kluyver, J. C.: On certain series of Mr. Hardy, Quart. J. Pure Appl. Math. 50 (1927), 185–192.30. Knuth, D. E.: The Art of Computer Programming, Vol. 3, Addison-Wesley, Englewood Cliffs,

1973.31. Lehmer, D. H.: The sum of like powers of the zeros of the Riemann zeta function, Math.

Comput. 50 (1988), 265–273.32. Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis, J. Number

Theory 65 (1997), 325–333.33. Li, X.-J.: A formula for the Dedekind ξ -function of an imaginary quadratic field, J. Math. Anal.

Appl. 260 (2001), 404–420.34. Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions, Illinois J. Math. 48 (2004),

491–503.35. Luke, Y. L.: The Special Functions and Their Approximations, Academic Press, New York,

1969.36. Maslanka, K.: Effective method of computing Li’s coefficients and their properties, submitted

to Experimental Math. (2004), http://nac.oa.uj.edu.pl/~maslanka37. Mestre, J.-F.: Formules explicites et minorations de conducteurs de variétés algébriques,

Compositio Math. 58 (1986), 209–232.38. Mitrovic, D.: The signs of some constants associated with the Riemann zeta function, Michigan

Math. J. 9 (1962), 395–397.


39. Nan-Yue, Z. and Williams, K. S.: Some results on the generalized Stieltjes constants, Analysis14 (1994), 147–162. In addition to the typographical errors pointed out in Appendix C, the firstterm on the right side of Equation (6.2) should read (logn a)/a, and in both Equations (6.5)and (6.11), Pn should appear in place of P1. On p. 148 in Eq. (1.4), fn(x) should be replacedby fn′ (x). On p. 157 of this reference, (7.2) should be replaced with (7.1) in the second line oftext from the bottom, and on p. 158 (7.2) should be replaced with (1.9) in the third line of text.

40. Neukirch, J.: Algebraic Number Theory, Springer, 1999.41. Odlyzko, A. M.: On the distribution of spacings between zeros of the zeta function, Math.

Comp. 48 (1987), 273–308; http://www.dtc.umn.edu/ odlyzko/42. Odlyzko, A. M.: Bounds for discriminants and related estimates for class numbers, regulators

and zeros of zeta functions: A survey of recent results, Sém. Théor. Nombres Bordeaux g (1989),1–15.

43. Omar, S.: Localization of the first zero of the Dedekind zeta function, Math. Comp. 70 (2001),1607–1616.

44. Rao, K. S., Berghe, G. V. and Krattenthaler, C.: An entry of Ramanujan on hypergeometricseries in his notebooks, J. Comput. Appl. Math. 173 (2005), 239–246.

45. Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsh. Preuss.Akad. Wiss. 671 (1859–1860).

46. Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function, 2nd edn, Oxford UniversityPress, Oxford, 1986.

47. Voros, A.: A sharpening of Li’s criterion for the Riemann hypothesis, arXiv:math.NT/0404213,2004.

48. Weil, A.: Basic Number Theory, Springer, New York, 1967.


Lifshits Tails Caused by Anisotropic Decay:The Emergence of a Quantum-Classical Regime

Dedicated to the memory of G. A. Mezincescu (1943–2001)

WERNER KIRSCH1 and SIMONE WARZEL2,�1Institut für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany.e-mail: [email protected] für Theoretische Physik, Universität Erlangen–Nürnberg, Staudtstr. 7,91058 Erlangen, Germany.

(Received: 12 January 2005; accepted: 2 July 2005)

Abstract. We investigate Lifshits-tail behaviour of the integrated density of states for a wide classof Schrödinger operators with positive random potentials. The setting includes alloy-type and Pois-sonian random potentials. The considered (single-site) impurity potentials f : R

d → [0, ∞[ decay atinfinity in an anisotropic way, for example, f (x1, x2) ∼ (|x1|α1 + |x2|α2 )−1 as |(x1, x2)| → ∞. Asis expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponentd/2 if both α1 and α2 are big enough, and there is a so-called classical regime with Lifshits exponentdepending on α1 and α2 if both are small. In addition to this we find two new regimes where theLifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transitionlines between these regimes depend in a nontrivial way on α1 and α2 simultaneously.

Mathematics Subject Classifications (2000):

Key words: random Schrödinger operators, integrated density of states, Lifshits tails.

1. Introduction

The integrated density of states N : R → [0, ∞[ is an important basic quantityin the theory of disordered electronic systems [3, 14, 24, 25, 34, 41, 42]. Roughlyspeaking, N(E) describes the number of energy levels below a given energy E perunit volume (see (15) below for a precise definition). A characteristic feature ofdisordered systems is the behaviour of N near band edges. It was first studied byLifshits [27]. He gave convincing physical arguments that the polynomial decrease

log N(E) ∼ log(E − E0)d2 as E ↓ E0 (1)

known as van-Hove singularity (see [22] for a rigorous proof) near a band edge E0

of an ideal periodic system in d space dimensions is replaced by an exponential

� Present address: Department of Physics, Princeton University, Jadwin Hall, Princeton,NJ 08544, U.S.A. E-mail: [email protected]

258 WERNER KIRSCH AND SIMONE WARZEL

decrease in a disordered system. In his honour, this decrease is known as Lifshitssingularity or Lifshits tail and typically given by

log N(E) ∼ log e−C(E−E0)−η

as E ↓ E0, (2)

where η > 0 is called the Lifshits exponent and C > 0 is some constant.The first rigorous proof [5] (see [32]) of Lifshits tails (in the sense that (2) holds)

concerns the bottom E0 of the energy spectrum of a continuum model involvinga Poissonian random potential

Vω(x) :=∑

j

f (x − ξω,j ), (3)

where ξω,j ∈ Rd are Poisson distributed points and f : R

d → [0, ∞[ is a non-negative impurity potential. Donsker and Varadhan [5] particularly showed that theLifshits exponent is universally given by η = d/2 in case

0 � f (x) � f0(1 + |x|)−α with some α > d + 2 and some f0 > 0. (4)

It was Pastur [33] who proved that the Lifshits exponent changes to η = d/(α −d)

if

fu(1 + |x|)−α � f (x)

� f0(1 + |x|)−α with some d < α < d + 2

and some fu, f0 > 0. (5)

This change from a universal Lifshits exponent to a nonuniversal one, which de-pends on the decay exponent α of f , may be heuristically explained in terms of acompetition of the kinetic and the potential energy of the underlying one-particleSchrödinger operator. In the first case (η = d/2) the quantum mechanical kineticenergy has a crucial influence on the (first order) asymptotics of N . The Lifshits tailis then said to have a quantum character. In the other case it is said to have a classi-cal character since then the (classical) potential energy determines the asymptoticsof N . For details, see for example [24, 26, 34].

Analogous results have been obtained for other random potentials. For example,in case of an alloy-type random potential

Vω(x) :=∑

j∈Zd

qω,jf (x − j) (6)

which is given in terms of independent identically distributed random variables qω,j

and an impurity potential f : Rd → [0, ∞[, the Lifshits tails at the lowest band

edge E0 have been investigated by [19, 21, 29]. Similarly to the Poissonian casethe authors of [21, 29] consider f as in (4) and (5) and detect a quantum and aclassical regime for which the Lifshits exponent equals

η =

d

2in case (4): d + 2 < α,

d

α − din case (5): d < α < d + 2

= max

{d

2,

d/α

1 − d/α

}. (7)

LIFSHITS TAILS CAUSED BY ANISOTROPIC DECAY 259

In fact they do not obtain the asymptotics (2) on a logarithmic scale but onlydouble-logarithmic asymptotics (confer (16) below). (See also [40] for an alter-native proof of this double-logarithmic asymptotics in case of alloy-type and Pois-sonian random potentials.)

Our main point is to generalise these results on the Lifshits exponent to impuritypotentials f that decay in an anisotropic way at infinity (confer (8) below). Inaddition we are able to handle a wide class of random potentials given in termsof random Borel measures which include among further interesting examples boththe case of alloy-type potentials and Poisson potential. Thus the same proof worksfor these two most important cases.

In our opinion it is interesting to explore the transition between quantum andclassical Lifshits behaviour in such models from both a mathematical and a phys-ical point of view. The interesting cases are those for which f decays fast enoughin some directions to ensure a quantum character while it decays slowly in theother direction so that the expected character there is the classical one. In thefollowing we give a complete picture of the classical and the quantum regime ofthe integrated density of states as well as of the emerging mixed quantum-classicalregime. We found it remarkable that the borderline between the quantum and clas-sical behaviour caused by the decay of f in a certain direction is not determinedby the corresponding decay exponent of these directions alone, but depends also ina nontrivial way on the decay in the other directions.

A second motivation for this paper came from investigations of the Lifshits tailsin a constant magnetic field in three space dimensions [10, 26, 43]. In contrast tothe two-dimensional situation [2, 8, 9, 11, 12, 43], the magnetic field introduces ananisotropy in R

3, such that it is quite natural to look at f which are anisotropic aswell. In fact, in the three-dimensional magnetic case a quantum-classical regimehas already been shown to occur for certain f with isotropic decay [26, 43]. Thepresent paper will contribute to a better understanding of these results.

The results mentioned above as well as the results in this paper concern Lifshitstails at the bottom of the spectrum. In accordance with Lifshits’ heuristics, theintegrated density of states should behave in a similar way at other edges of thespectrum. Such internal Lifshits tails were proven in [15, 16, 23, 28, 30, 38].

2. Basic Quantities and Main Result

2.1. RANDOM POTENTIALS

We consider random potentials

V : � × Rd −→ [0, ∞[,

(ω, x) �−→ Vω(x) :=∫

Rd

f (x − y)µω(dy),(8)

which are given in terms of a random Borel measure µ: � → M(Rd), ω �→ µω,and an impurity potential f : R

d → [0, ∞[. We recall from [6, 13, 39] that a ran-


dom Borel measure is a measurable mapping from a probability space (�, A, P)

into the set of Borel measures (M(Rd), B(M)), that is, the set of positive, locally-finite measures on R

d . Here B(M) denotes the Borel σ -algebraof M(Rd), that is, the smallest σ -algebra rendering the mappings M(Rd) � ν �→ν() measurable for all bounded Borel sets ∈ B(Rd).

The following assumptions on µ are supposed to be valid throughout the paper.

ASSUMPTION 2.1. The random Borel measure µ: � → M(Rd), ω �→ µω isdefined on some complete probability space (�, A, P). We suppose that:

(i) µ is Zd-stationary.

(ii) There exists a partition of Rd = ⋃

j∈Zd j into disjoint unit cubes j =0 + j centred at the sites of the lattice Z

d such that the random variables(µ((j))j∈J are stochastically independent for any finite collection J ⊂ Z

d

of Borel sets (j) ⊂ j .(iii) The intensity measure µ: B(Rd) → [0, ∞[, which is given by

µ() := E[µ()] (9)

in terms of the probabilistic expectation E[·] := ∫�(·)P(dω), is a Borel

measure which does not vanish identically µ = 0.(iv) There is some constant κ > 0 such that P{ω ∈ �: µω(0) ∈ [0, ε[} � εκ for

small enough ε > 0.

Remark 2.2. Assumption 2.1(i) implies that the intensity measure µ isZ

d-periodic. Assumption 2.1(iii) is thus equivalent to the existence of the firstmoment E[µ(0)] < ∞ of the random variable µ(0): ω �→ µω(0). Moreover,we emphasis that the unit cubes (j ) introduced in Assumption 2.1(ii) are neitheropen nor closed.

We recall from [13, 39, 6] that Zd-stationarity of µ requires the group (Tj )j∈Zd

of lattice translations, which is defined on M(Rd) by (Tjν)() := ν(+ j) for all ∈ B(Rd) and all j ∈ Z

d , to be probability preserving in the sense that

P {TjM} = P {M} (10)

for all M ∈ B(M) and all j ∈ Zd . Here we have introduced the notation P {M} :=

P{ω ∈ � : µω ∈ M} for the induced probability measure on (M(Rd), B(M)). Toensure the (Zd-)ergodicity of the random potential V , it is useful to know thatunder the assumptions made above, (Tj ) is a group of mixing (hence ergodic)transformations on the probability space (M(Rd), B(M), P ).

LEMMA 2.3. Assumptions 2.1(i) and 2.1(ii) imply that µ is mixing in the sensethat

lim|j |→∞

P {TjM ∩ M ′} = P {M}P {M ′} (11)

for all M, M ′ ∈ B(M).


Proof. See Appendix A. �The considered impurity potentials f : R

d → [0, ∞[ comprise a large class offunctions with anisotropic decay. More precisely, we decompose the configurationspace R

d = Rd1×· · ·×R

dm into m ∈ N subspaces with dimensions d1, . . . , dm ∈ N.Accordingly, we will write x = (x1, . . . , xm) ∈ R

d , where xk ∈ Rdk and k ∈

{1, . . . , m}. Denoting by |xk| := maxi∈{1,...,dk} |(xk)i | the maximum norm on Rdk ,

our precise assumptions on f are as follows.

ASSUMPTION 2.4. The impurity potential f : Rd → [0, ∞[ is positive, strictly

positive on some nonempty open set and satisfies:

(i) The Birman–Solomyak condition∑

j∈Zd (∫0

|f (x − j)|p dx)1/p < ∞ withp = 2 if d ∈ {1, 2, 3} and p > d/2 if d � 4.

(ii) There exist constants α1, . . . , αm ∈ [0, ∞] and 0 < fu, f0 < ∞ such that

fu∑mk=1 |xk|αk

�∫

0

f (y − x) dy, f (x) � f0∑mk=1 |xk|αk

(12)

for all x = (x1, . . . , xm) ∈ Rd with large enough values of their maximum

norm |x| = max{|x1|, . . . , |xm|}.Remark 2.5. In order to simultaneously treat the case αk = ∞ for some (or all)

k ∈ {1, . . . , m}, we adopt the conventions |xk|∞ := ∞ for |xk| > 0 and 1/∞ := 0.An example for such a situation is given by f with compact support in thexk-direction.

2.2. EXAMPLES

The setting in Subsection 2.1 covers a huge class of random potentials which arewidely encountered in the literature on random Schrödinger operators [3, 14, 34,41]. In this subsection we list prominent examples, some of which have alreadybeen (informally) introduced in Section 1.

From the physical point of view, it is natural to consider integer-valued randomBorel measures ν = ∑

j kj δxj, also known as point processes [6]. Here each kj is

an integer-valued random variable and the distinct points (xj ) indexing the atoms,equivalently the Dirac measure δ, form a countable (random) set with at mostfinitely many xj in any bounded Borel set. In fact, interpreting (xj ) as the (random)positions of impurities in a disordered solid justifies the name ‘impurity potential’for f in (8).

Two examples of point processes satisfying Assumptions 2.1(i)–2.1(iii) are:

(P) The generalised Poisson measure ν = ∑j δξj

with some nonzero Zd-periodic

Borel intensity measure ν. The Poisson measure is uniquely characterisedby requiring that the random variables ν((1)), . . . , ν((n)) are stochastically


independent for any collection of disjoint Borel sets (1), . . . , (n) ∈ B(Rd)

and that each ν() is distributed according to Poisson’s law

P{ω ∈ � : νω() = k} = (ν())k

k! exp[−ν()], k ∈ N0 (13)

for any bounded ∈ B(Rd). The case ν() = || corresponds to the usualPoisson process with parameter > 0.

(D) The displacement measure ν = ∑j∈Zd δj+dj

. Here the random variablesdj ∈ 0 are independent and identically distributed over the unit cube. Thecase dj = 0 corresponds to the (nonrandom) periodic point measure ν =∑

j∈Zd δj .

Any (generalised) Poisson measure (P) also satisfies Assumption 2.1(iv). Itgives rise to the (generalised) Poissonian random potential (3). Unfortunately, As-sumption 2.1(iv) is never satisfied for any displacement measure (D). However,a corresponding compound point process ν = ∑

j∈Zd qj δxjwill satisfy Assump-

tion 2.1(iv) under suitable conditions on the random variables (qj ). In order to sat-isfy Assumption 2.1(iii), we take (qj )j∈Zd independent and identically distributed,positive random variables with 0 < E[q0] < ∞.

Two examples of such compound point processes, for which Assumptions2.1(i)–2.1(iv) hold, are:

(P′) The compound (generalised) Poisson measure ν = ∑j qj δξj

with (ξj ) asin (P).

(D′) The compound displacement measure ν = ∑j∈Zd qj δj+dj

with dj as in (D).Assumption 2.1(iv) requires P{ω ∈ � : qω,0 ∈ [0, ε[} � εκ for small enoughε > 0 and some κ > 0. The case dj = 0 gives the alloy-type measureν = ∑

j∈Zd qj δj associated with the alloy-type random potential (6).

Remark 2.6. We note that in case (P′) there are no further requirements on (qj ).Moreover, our results in Subsection 2.4 below also apply to alloy-type randompotentials (6) with bounded below random variables (qj ), not only positive ones.This follows from the fact that one may add x �→ ∑

j∈Zd qminf (x − j) to theperiodic background potential Uper (confer (14) and Assumption 2.7 below).

2.3. RANDOM SCHÖDINGER OPERATORS AND THEIR INTEGRATED DENSITY

OF STATES

For any of the above defined random potentials V , we study the correspondingrandom Schrödinger operator, which is informally given by the second order dif-ferential operator

H(Vω) := −� + Uper + Vω (14)


on the Hilbert space L2(Rd) of complex-valued, square-integrable functions on Rd .

Thereby the periodic background potential Uper (acting in (14) as a multiplicationoperator) is required to satisfy the following

ASSUMPTION 2.7. The background potential Uper: Rd → R is Z

d-periodic andUper ∈ Lp

loc(Rd) for some p > d.

Assumptions 2.1 and 2.4 particularly imply [3, Cor. V.3.4] that Vω ∈ Lp

loc(Rd)

for P-almost all ω ∈ � with the same p as in Assumption 2.4(ii). Together withAssumption 2.7 this ensures [20] that H(Vω) is essentially self-adjoint on the spaceC∞

c (Rd) of complex-valued, arbitrarily often differentiable functions with compactsupport for P-almost all ω ∈ �. Since V is Z

d-ergodic (confer Lemma 2.3),the spectrum of H(Vω) coincides with a nonrandom set for P-almost all ω ∈ �

[18, Theorem 1].For any d-dimensional open cuboid ⊂ R

d , the restriction of (14) to C∞c ()

defines a self-adjoint operator HD (Vω) on L2(), which corresponds to taking

Dirichlet boundary conditions [35]. It is bounded below and has purely discretespectrum with eigenvalues λ0(H

D (Vω) < λ1(H

D (Vω) � λ2(H

D (Vω) � · · · or-

dered by magnitude and repeated according to their multiplicity. Our main quantityof interest, the integrated density of states, is then defined as the infinite-volumelimit

N(E) := lim||→∞1

||#{n ∈ N0 : λn(HD (Vω)) < E}. (15)

More precisely, thanks to the Zd-ergodicity of the random potential there is a set

�0 ∈ A of full probability, P(�0) = 1, and a nonrandom unbounded distributionfunction N : R → [0, ∞[ such that (15) holds for all ω ∈ �0 and all continuitypoints E ∈ R of N . The set of growth points of N coincides with the almost-surespectrum of H(Vω), confer [3, 14, 34].

2.4. LIFSHITS TAILS

The main result of the present paper generalises the result (7) of [21, 29] onthe Lifshits exponent for alloy-type random potentials with isotropically decay-ing impurity potential f to the case of anisotropic decay and more general ran-dom potentials (8). We note that isotropic decay corresponds to taking m = 1 inAssumption 2.4 or, what is the same, α := αk for all k ∈ {1, . . . , m}.THEOREM 2.8. Let H(Vω) be a random Schrödinger operator (14) with randompotential (8) satisfying Assumptions 2.1 and 2.4, and a periodic background poten-tial satisfying Assumption 2.7. Then its integrated density of states N drops downto zero exponentially near E0 := inf spec H(0) with Lifshits exponent given by

η := limE↓E0

log|log N(E)||log(E − E0)| =

m∑

k=1

max

{dk

2,

γk

1 − γ

}, (16)

where γk := dk/αk and γ := ∑mk=1 γk.


Remarks 2.9. (i) As a by-product, it turns out that the infimum of the almost-sure spectrum of H(Vω) coincides with that of H(0) = −� + Uper.

(ii) Thanks to the convention 0 = dk/∞(= γk), Theorem 2.8 remains valid ifαk = ∞ for some (or all) k ∈ {1, . . . , m}, confer Remark 2.5.

(iii) Assumption 2.7 on the local singularities of Uper is slightly more restrictivethan the one in [21, 29]. It is tailored to ensure certain regularity properties ofthe ground-state eigenfunction of H(0). As can be inferred from Subsection 3.1below, we may relax Assumption 2.7 and require only p > d/2 (as in [21, 29]) inthe interior of the unit cube and thus allow for Coulomb singularities there.

(iv) Even in the isotropic situation m = 1 Assumption 2.4 covers slightly moreimpurity potentials than in [21, 29], since we allow f to have zeros at arbitrarylarge distance from the origin.

(v) An inspection of the proof below shows that we prove a slightly betterestimate than the double logarithmic asymptotics given in (16). In particular, ifthe measure µω has an atom at zero, more exactly if P{ω ∈ � : µω(0) = 0} > 0,then we actually prove

−C(E − E0)−η � log N(E) � −C ′(E − E0)

−η (17)

for small E − E0. This is not quite the logarithmic behaviour (2) of N since theconstants C > 0 and C ′ > 0 do not agree. Note that µω has an atom at zerofor any generalized Poisson measure (P) as well as for a compound displacementmeasure (D′) if P{ω ∈ � : qω,0(ω) = 0} > 0.

For an illustration and interpretation of Theorem 2.8 we consider the specialcase m = 2. The right-hand side of (16) then suggests to distinguish the followingthree cases:

Quantum regime:

d1

2� γ1

1 − γand

d2

2� γ2

1 − γ. (qm)

Quantum-classical regime:

d1

2� γ1

1 − γand

d2

2<

γ2

1 − γ(qm/cl)

or

d1

2<

γ1

1 − γand

d2

2� γ2

1 − γ. (cl/qm)

Classical regime:

d1

2<

γ1

1 − γand

d2

2<

γ2

1 − γ. (cl)


In comparison to the result (7) for m = 1 the main finding of this paper is theemergence of a regime corresponding to mixed quantum and classical characterof the Lifshits tail. A remarkable fact about the Lifshits exponent (16) is that thedirections k ∈ {1, 2} related to the anisotropy do not show up separately as onemight expect naively. In particular, the transition from a quantum to a classicalregime for the xk-direction does not occur if dk/2 = γk/(1 − γk), but rather ifdk/2 = γk/(1 − γ ). This intriguing intertwining of directions through γ maybe interpreted in terms of the marginal impurity potentials f (1) and f (2) definedin (24) and (25) below. In fact, when writing γ2/(1 − γ ) = d2/(α2(1 − γ1) − d2)

and identifying α2(1 − γ1) as the decay exponent of f (2) by Lemma 3.4 below,it is clear that f (2) serves as an effective potential for the x2-direction as far asthe quantum-classical transition is concerned. In analogy, f (1) serves as the effec-tive potential for the x1-direction. Heuristic arguments for the importance of themarginal potentials in the presence of an anisotropy can be found in [26].

3. Basic Inequalities and Auxiliary Results

In order to keep our notation as transparent as possible, we will additionally sup-pose that

E0 = 0 and m = 2 (18)

throughout the subsequent proof of Theorem 2.8. In fact, the first assumption canalways be achieved by adding a constant to H(0).

The strategy of the proof is roughly the same as in [21, 29], which in turnis based on [19, 37]. We use bounds on the integrated density of states N andsubsequently employ the Rayleigh–Ritz principle and Temple’s inequality [35]to estimate the occurring ground-state energies from above and below. The basicidea to construct the bounds on N is to partition the configuration space R

d intocongruent domains and employ some bracketing technique for H(Vω). The moststraightforward of these techniques is Dirichlet or Neumann bracketing. However,to apply Temple’s inequality to the arising Neumann ground-state energy, the au-thors of [21] required that Uper is reflection invariant. To get rid of this additionalassumption, Mezincescu [29] suggested an alternative upper bound on N which isbased on a bracketing technique corresponding to certain Robin (mixed) bound-ary conditions. In his honour, we will refer to these particular Robin boundaryconditions as Mezincescu boundary conditions.

3.1. MEZINCESCU BOUNDARY CONDITIONS AND BASIC INEQUALITIES

Assumption 2.7 on Uper implies [36, Theorem C.2.4] that there is a continuouslydifferentiable representative ψ : R

d →]0, ∞[ of the strictly positive ground-stateeigenfunction of H(0) = −� + Uper, which is L2-normalised on the unit cube 0,


∫

0

ψ(x)2 dx = 1. (19)

The function ψ is Zd-periodic, bounded from below by a strictly positive constant

and obeys H(0)ψ = E0ψ = 0.Subsequently, we denote by ⊂ R

d a d-dimensional, open cuboid which iscompatible with the lattice Z

d , that is, we suppose that it coincides with the interiorof the union of Z

d-translates of the closed unit cube. On the boundary ∂ of wedefine χ : ∂ → R as the negative of the outer normal derivative of log ψ ,

χ(x) := − 1

ψ(x)(n · ∇)ψ(x), x ∈ ∂. (20)

Since χ ∈ L∞(∂) is bounded, the sesquilinear form

(ϕ1, ϕ2) �−→∫

∇ϕ1(x) · ∇ϕ2(x) dx +∫

∂

χ(x)ϕ1(x)ϕ2(x) dx, (21)

with domain ϕ1, ϕ2 ∈ W 1,2() := {ϕ ∈ L2() : ∇jϕ ∈ L2() for all j ∈{1, . . . , d}}, is symmetric, closed and lower bounded, and thus uniquely definesa self-adjoint operator −�

χ

=: Hχ

(0) − Uper on L2(). In fact, the conditionχ ∈ L∞(∂) guarantees that boundary term in (21) is form-bounded with boundzero relative to the first term, which is just the quadratic form corresponding tothe (negative) Neumann Laplacian. Consequently [35, Theorem XIII.68], both theRobin Laplacian −�

χ

as well as Hχ

(Vω) := −�χ

+ Uper + Vω, defined as a formsum on W 1,2() ⊂ L2(), have compact resolvents. Since H

χ

(Vω) generates apositivity preserving semigroup, its ground-state is simple and comes with a strictlypositive eigenfunction [35, Theorem XIII.43].

Remarks 3.1. (i) In the boundary term in (21) we took the liberty to denote thetrace of ϕj ∈ W 1,2() on ∂ again by ϕj .

(ii) Partial integration shows that the quadratic form (21) corresponds to impos-ing Robin boundary conditions (n · ∇ +χ)ψ |∂ = 0 on functions ψ in the domainof the Laplacian on L2(). Obviously, Neumann boundary conditions correspondto the special case χ = 0. With the present choice (20) of χ they arise if Uper = 0such that ψ = 1 or, more generally, if Uper is reflection invariant (as was supposedin [21]).

(iii) Denoting by λ0(Hχ

(Vω)) < λ1(Hχ

(Vω)) � λ2(Hχ

(Vω)) � · · · the eigen-values of H

χ

(Vω), the eigenvalue-counting function

N(E;Hχ

(Vω)) := #{n ∈ N0 : λn(Hχ

(Vω)) < E} (22)

is well-defined for all ω ∈ � and all energies E ∈ R. If Uper is bounded from below,it follows from [31, Theorem 1.3] and (15) that N(E) = lim||→∞ ||−1N(E;H

χ

(Vω)). We also refer to [31] for proofs of some of the above-mentioned proper-ties of the Robin Laplacian.


One important point about the Mezincescu boundary conditions (20) is that therestriction of ψ to continues to be the ground-state eigenfunction of H

χ

(0) witheigenvalue λ0(H

χ

(0)) = E0 = 0. This follows from the fact that ψ satisfies theeigenvalue equation, the boundary conditions and that ψ is strictly positive.

Our proof of Theorem 2.8 is based on the following sandwiching bound on theintegrated density of states.

PROPOSITION 3.2. Let ⊂ Rd be a d-dimensional open cuboid, which is

compatible with the lattice Zd . Then the integrated density of states N obeys

||−1P{ω ∈ � : λ0(H

D (Vω)) < E}

� N(E) � ||−1N(E;Hχ

(0))P{ω ∈ � : λ0(Hχ

(Vω)) < E} (23)

for all energies E ∈ R.Proof. For the lower bound on N , see [19, Eqs. (4) and (21)] or [21, Eq. (2)].

The upper bound follows from [29, Eq. (29)]. �Remark 3.3. Since the bracketing [29, Prop. 1], [3, Problem I.7.19] applies to

Robin boundary conditions with more general real χ ∈ L∞(∂) than the onedefined in (20), the same is true for the upper bound in (23).

3.2. ELEMENTARY FACTS ABOUT MARGINAL IMPURITY POTENTIALS

Key quantities in our proof of Theorem 2.8 are the marginal impurity potentialsf (1): R

d1 → [0, ∞[ and f (2): Rd2 → [0, ∞[ for the x1- and x2-direction, respec-

tively. For the given f ∈ L1(Rd) they are defined as follows

f (1)(x1) :=∫

Rd2

f (x1, x2) dx2, (24)

f (2)(x2) :=∫

Rd1

f (x1, x2) dx1. (25)

The aim of this subsection is to collect properties of f (2). Since f (1) results fromf (2) by exchanging the role of x1 and x2, analogous properties apply to f (1).

LEMMA 3.4. Assumption 2.4 with m = 2 implies that there exist two constants0 < f1, f2 < ∞ such that

f1

|x2|α2(1−γ1)�

∫

|y2|< 12

f (2)(y2 − x2) dy2, f (2)(x2) � f2

|x2|α2(1−γ1)(26)

for large enough |x2| > 0.Proof. The lemma follows by elementary integration. In doing so, one may

replace the maximum norm | · | by the equivalent Euclidean 2-norm in both (12)and (26). �


LEMMA 3.5. Assumption 2.4 with m = 2 implies that there exists some constant0 < f3 < ∞ such that

∫

|x2|>L

f (2)(x2) dx2 � f3L−α2(1−γ ) (27)

for sufficiently large L > 0.Proof. By Lemma 3.4 we have

∫|x2|>L

f (2)(x2) dx2 � f2∫|x2|>L

|x2|−α2(1−γ1) dx2

for sufficiently large L > 0. The assertion follows by elementary integration andthe fact that α2(1 − γ1) − d2 = α2(1 − γ ). �

Remark 3.6. One consequence of Lemma 3.5, which will be useful below, isthe following inequality

sup|y2|�L/2

∫

|x2|>Lβ

f (2)(x2 − y2) dx2 � f3

(2

Lβ

)α2(1−γ )

(28)

valid for all β � 1 and sufficiently large L > 1. It is obtained by observing that theintegral in (28) equals

∫

|x2+j2|>Lβ

f (2)(x2) dx2 �∫

|x2|�Lβ/2f (2)(x2) dx2. (29)

Here the last inequality results from the triangle inequality |x2 + y2| � |x2| + |y2|and the fact that |y2|L/2 � Lβ/2.

4. Upper Bound

For an asymptotic evaluation of the upper bound in Proposition 3.2 for small en-ergies E we distinguish the three regimes defined below Theorem 2.8: quantum,quantum-classical and classical.

4.1. REGULARISATION OF RANDOM BOREL MEASURE

In all of the above mentioned cases it will be necessary to regularise the givenrandom Borel measure µ by introducing a cut off. For this purpose we define aregularised random Borel measure µ(h): � × B(Rd) → [0, ∞[ with parameterh > 0 by µ(h)

ω () := ∑j∈Zd µ(h)

ω ( ∩ j) where

µ(h)ω ( ∩ j) :=

µω( ∩ j) µω(j) � h,

hµω( ∩ j)

µω(j)otherwise

(30)

for all ∈ B(Rd) and all ω ∈ �.


Remark 4.1. Since µ(h)ω (∅) = 0 and µ(h)

ω (⋃

n (n)) = ∑n µ(h)

ω ((n)) for anycollection of disjoint (n) ∈ B(Rd), each realization µ(h)

ω is indeed a measureon the Borel sets B(Rd). It is locally finite and hence a Borel measure, becauseµ(h)

ω (j ) � h for all j ∈ Zd and all ω ∈ �.

For future reference we collect some properties of µ(h).

LEMMA 4.2. Let h > 0. Then the following three assertions hold true:

(i) µ(h)ω () � min{µω(), h#{j ∈ Z

d : ∩ j = ∅}} for all ∈ B(Rd) andall ω ∈ �.

(ii) The intensity measure µ(h): B(Rd) → [0, ∞[ given by µ(h)() :=E[µ(h)()] is a Borel measure which is Z

d-periodic and obeys µ(h)(0) > 0.(iii) The random variables (µ(h)(j ))j∈Zd are independent and identically dis-

tributed.

Proof. The first part of the first assertion is immediate. The other part followsfrom the monotonicity µω(∩j) � µω(j) � h for all ∈ B(Rd), j ∈ Z

d andall ω ∈ �. The claimed Z

d-periodicity of the intensity measure is traced back tothe Z

d-stationarity of µ. The inequality in the second assertion holds, since µω(0)

is not identical zero for P-almost all ω ∈ � (confer Assumption 2.1). The thirdassertion follows from the corresponding property of µ (confer Assumption 2.1). �

4.2. QUANTUM REGIME

Throughout this subsection we suppose that (qm) holds. Assumption 2.4 on theimpurity potential requires the existence of some constant fu > 0 and some Borelset F ∈ B(Rd) with |F | > 0 such that

f � fuχF . (31)

Without loss of generality, we will additionally suppose that F ⊂ 0. We startby constructing a lower bound on the lowest Mezincescu eigenvalue λ0(H

χ

(Vω))

showing up in the right-hand side of (23) when choosing the interior of the closure

:=⋃

|j |<L

j

int

(32)

of unit cubes, which are at most at a distance L > 1 from the origin. By construc-tion, the cube is open and compatible with the lattice.

4.2.1. Lower Bound on the Lowest Mezincescu Eigenvalue

From Lemma 4.2(i) and (31) we conclude that the potential Vω,h: Rd → [0, ∞[

given by

Vω,h(x) := fu

∫

Rd

χF (x − y)µ(h)ω (dy) = fuµ

(h)ω (x − F) (33)


in terms of the regularised Borel measure µ(h)ω , provides a lower bound on Vω for

every h > 0 and ω ∈ �. The fact that the pointwise difference x − F is containedin a cube, which consists of (at most) 3d unit cubes, together with Lemma 4.2(i)implies the estimate

Vω,h(x) � 3dfuh (34)

for all ω ∈ � and all x ∈ Rd . Taking h small enough thus ensures that the maxi-

mum of the potential Vω,h is smaller than the energy difference of the lowest and thefirst eigenvalue of H

χ

(0). This enables one to make use of Temple’s inequality toobtain a lower bound on the lowest Mezincescu eigenvalue in the quantum regime.

PROPOSITION 4.3. Let denote the open cube (32). Moreover, let h := (r0L)−2

with r0 > 0. Then the lowest eigenvalue of Hχ

(Vω,h) is bounded from belowaccording to

λ0(Hχ

(Vω,h)) � 1

2||∫

Vω,h(x)ψ(x)2 dx (35)

for all ω ∈ �, all L > 1 and large enough r0 > 0. [Recall the definition of ψ atthe beginning of Subsection 3.1.]

Proof. By construction ψL := ||−1/2ψ ∈ L2() is the normalised ground-state eigenfunction of H

χ

(0) which satisfies Hχ

(0)ψL = 0. Choosing this functionas the variational function in Temple’s inequality [35, Theorem XIII.5] yields thelower bound

λ0(Hχ

(Vω,h)) � 〈ψL, Vω,hψL〉 − 〈Vω,hψL, Vω,hψL〉λ1(H

χ

(0)) − 〈ψL, Vω,hψL〉 (36)

provided the denominator in (36) is strictly positive. To check this we note that[29, Prop. 4] implies that there is some constant c0 > 0 such that

λ1(Hχ

(0)) = λ1(Hχ

(0)) − λ0(Hχ

(0)) � 2c0L−2 (37)

for all L > 1. Moreover, we estimate 〈ψL, Vω,hψL〉 � 3dfuh � c0L−2 for large

enough r0 > 0. To bound the numerator in (50) from above, we use the inequality〈Vω,hψL, Vω,hψL〉 � 〈ψL, Vω,hψL〉3dfuh � 〈ψL, Vω,hψL〉c0/(2L2) valid for largeenough r0 > 0. �

We proceed by constructing a lower bound on the right-hand side of (36). Forthis purpose we define the cube

:=⋃

|j |<L−1

j (38)

which is contained in the cube defined in (32). In fact it is one layer of unit cubessmaller than .


LEMMA 4.4. There exists a constant 0 < c1 < ∞ (which is independent of ω, L

and h) such that

1

||∫

Vω,h(x)ψ(x)2 dx � c1h

||#{j ∈ Zd ∩ : µω(j) � h} (39)

for all ω ∈ �, all L > 1 and all h > 0.Proof. Pulling out the strictly positive infimum of ψ2 and using its Z

d-periodici-ty, we estimate

∫

Vω,h(x)ψ(x)2 dx � infz∈0

ψ(z)2fu

∫

Rd

| ∩ (F + y)|µ(h)ω (dy)

� infz∈0

ψ(z)2fu|F |µ(h)ω () (40)

by omitting positive terms and using Fubini’s theorem together with the fact thatF ⊂ 0. The proof is completed with the help of the inequality

µ(h)ω () =

∑

j∈Zd∩

min{h, µω(j)} � h#{j ∈ Zd ∩ : µω(j) � h} (41)

and || � 3d || valid for all L > 1. �

4.2.2. Proof of Theorem 2.8 – First Part: Quantum Regime

We fix r0 > 0 large enough to ensure the validity of (35) in Proposition 4.3. For agiven energy E > 0 we then pick

L :=(

c1

4r20E

)1/2

, (42)

where the constant c1 has been fixed in Lemma 4.4. Finally, we choose the cube

from (32) and set h := (r0L)−2. Proposition 4.3 and (39) yield the estimate

P{ω ∈ � : λ0(Hχ

(Vω)) < E}� P

{ω ∈ � : #{j ∈ Z

d ∩ : µω(j) � h} <2E

c1h||

}

= P

{ω ∈ � : #{j ∈ Z

d ∩ : µω(j) < h} >||2

}. (43)

Here the last equality uses the fact that h = 4E/c1. In case µ(j) > h, that is,for sufficiently small E, the right-hand side is the probability of a large deviationevent [7]. Consequently (confer [21, Prop. 4]), there exists a constant 0 < c2 < ∞,such that (43) is estimated from above by

exp[−c2||] � exp[−c2nuLd] = exp[−c3E

−d/2]. (44)


Here the inequality follows from the estimate || � nuLd for some constant

nu > 0 and all L > 2. The existence of a constant c3 > 0 ensuring the validityof the last equality follows from (42). Inserting this estimate in the right-hand sideof (23) completes the first part of the proof of Theorem 2.8 for the quantum-classical regime, since the pre-factor in the upper bound in Proposition 3.2 isnegligible.

4.3. QUANTUM-CLASSICAL REGIME

Without loss of generality we suppose that (qm/cl) holds throughout this subsection,that is d1/2 � γ1/(1 − γ ) and d2/2 < γ2/(1 − γ ). We start by constructing alower bound on the lowest Mezincescu eigenvalue λ0(H

χ

(Vω)) showing up in theright-hand side of (23) when choosing

:=⋃

|j1|<L

(j1,0)

int

(45)

a cuboid with some L > 1. By construction it is open and compatible with thelattice.


From Lemma 4.2(i) we conclude that for every R > 0 and ω ∈ � the potentialVω,R: R

d → [0, ∞[ given by

Vω,R(x) :=∫

|y2|>R

f (x − y)µ(1)ω (dy) (46)

in terms of the regularised Borel measure µ(1)ω , provides a lower bound on Vω.

Therefore λ0(Hχ

(Vω)) � λ0(Hχ

(Vω,R)). It will be useful to collect some factsrelated to Vω,R.

LEMMA 4.5. Let R > 1 and define VR: Rd → [0, ∞[ by

VR(x) :=∑

j1∈Zd1

|j2|>R−1

supy∈j

f (x − y). (47)

Then the following three assertions hold true:

(i) Vω,R � VR for every ω ∈ �.(ii) VR is Z

d1-periodic with respect to translations in the x1-direction.(iii) There exists some constant c > 0 such that supx∈0

VR(x) � cR−α2(1−γ ) forlarge enough R > 1.


Proof. The first assertion follows from the inequalities

Vω,R(x) �∑

j1∈Zd1

|j2|>R−1

∫

j

f (x − y)µ(1)ω (dy) (48)

and µ(1)ω (j ) � 1 valid for all ω ∈ �. The second assertion holds true by defi-

nition. The third assertion derives from (12) and is the ‘summation’ analogue ofLemma 3.5. �

The cut-off R guarantees that the potential Vω,R does not exceed a certain value.In particular, taking R large enough ensures that this value is smaller than theenergy difference of the lowest and the first eigenvalue of H

χ

(0). This enablesone to make use of Temple’s inequality to obtain a lower bound on the lowestMezincescu eigenvalue in the quantum-classical regime.

PROPOSITION 4.6. Let denote the cuboid (45). Moreover, let R :=(r0L)2/α2(1−γ ) with r0 > 0. Then the lowest eigenvalue of H

χ

(Vω,R) is boundedfrom below according to

λ0(Hχ

(Vω,R)) � 1

2||∫

Vω,R(x)ψ(x)2 dx (49)

for all ω ∈ �, all L > 1 and large enough r0 > 0. [Recall the definition of ψ atthe beginning of Subsection 3.1.]

Proof. The proof parallels the one of Proposition 4.3. By construction ψL :=||−1/2ψ ∈ L2() is the normalised ground-state eigenfunction of H

χ

(0) whichsatisfies H

χ

(0)ψL = 0. Choosing this function as the variational function in Tem-ple’s inequality [35, Theorem XIII.5] yields the lower bound

λ0(Hχ

(Vω,R)) � 〈ψL, Vω,RψL〉 − 〈Vω,RψL, Vω,RψL〉λ1(H

χ

(0)) − 〈ψL, Vω,RψL〉 (50)

provided the denominator in (50) is strictly positive. To check this we note that asimple extension of [29, Prop. 4] from cubes to cuboids implies that there is someconstant c0 > 0 such that λ1(H

χ

(0)) = λ1(Hχ

(0)) − λ0(Hχ

(0)) � 2c0L−2 for all

L > 1. Moreover, using Lemma 4.5 and the definition of R we estimate

〈ψL, Vω,RψL〉 � 〈ψL, VRψL〉 =∫

0

VR(x)ψ(x)2 dx

� c(r0L)−2 � c0L−2 (51)

for large enough r0 > 0. To bound the numerator in (50) from above, we use theinequality 〈Vω,RψL, Vω,RψL〉 � 〈ψL, Vω,RψL〉 supx∈ VR(x). Lemma 4.5 ensuresthat supx∈ VR(x) = supx∈0

VR(x) and thus yields the bound

〈Vω,RψL, Vω,RψL〉 � 〈ψL, Vω,RψL〉c(r0L)−2 � 〈ψL, Vω,RψL〉c0

2L−2 (52)

for large enough r0 > 0. �


We proceed by constructing a lower bound on the right-hand side of (49). Forthis purpose we set

:=⋃

|j1|�L/8R<|j2|�2R

j (53)

a union of disjoint cuboids.

LEMMA 4.7. There exist two constants 0 < c2, c3 < ∞ (which are independentof ω, L and R) such that

∫

Vω,R(x)ψ(x)2 dx � c2

Rα2(1−γ1)µ(1)

ω () − c3||L−α1(1−γ ) (54)

for all ω ∈ � and large enough L > 1 and R > 1.

Remark 4.8. An important consequence of this lemma reads as follows. Thereexists some constant nu > 0 such that the number of lattice points in is estimatedfrom below by || � nu||Rd2 for all L > 1 and R > 1 and some constant nu > 0.Therefore ||/(||Rα2(1−γ1)) � nu/R

α2(1−γ ). Choosing R = (r0L)2/α2(1−γ ) as inProposition 4.6, we thus arrive at the lower bound

1

||∫

Vω,R(x)ψ(x)2 dx � c2nu

(r0L)2

1

||∑

j∈Zd∩

µ(1)ω (j ) − c3L

−α1(1−γ ) (55)

valid for all r0 > 0 and large enough L > 1.

Proof of Lemma 4.7. Pulling out the strictly positive infimum of ψ2 and usingits Z

d-periodicity, we estimate∫

Vω,R(x)ψ(x)2 dx � infz∈0

ψ(z)2∫

Vω,R(x) dx

� infz∈0

ψ(z)2∫

(∫

f (x − y) dx

)µ(1)

ω (dy) (56)

by omitting positive terms and using Fubini’s theorem. The inner integral in the lastline is estimated from below with the help of Lemma 3.4 in terms of the marginalimpurity potential f (2) (recall definition (25)) according to

∫

f (x − y) dx =∫

|x2|< 12

f (2)(x2 − y2) dx2 −∑

|k1|�L

∫

(k1,0)

f (x − y) dx

� f1

(2R + 1)α2(1−γ1)−

∑

|k1|�L

∫

0

f (x + (k1, 0) − y) dx (57)

for all |y2| � 2R + 1 and large enough R > 0. The first term on the right-handside yields the first term on the right-hand side of (54). To estimate the remainder


we decompose the y-integration of the second term in (57) with respect to µ(1)ω and

use the fact that µ(1)ω (j ) � 1. This yields an estimate of the form

∫

(∫

0

g(x − y) dx

)µ(1)

ω (dy) �∑

j∈Zd∩

supy∈0

∫

0

g(x − y − j) dx

� 3d∑

|j1|�L/2j2∈Z

d2

∫

0

g(x − j) dx

= 3d∑

|j1|�L/4

∫

|x1|<1/2g(1)(x1 − j1) dx1 (58)

valid for all g ∈ L1(Rd). Here the second inequality holds for every L � 8 (sothat L/4 − L/8 � 1) and follows from enlargening the j2-summation and the factthat the pointwise difference 0 −0 is contained in the cube centred at the originand consisting of 3d unit cubes. The last equality uses the definition (24) for amarginal impurity potential. Substituting g(x) = f (x + (k1, 0)) in the above chainof inequalities, performing the k1-summation and enlargening the x1-integrationthus yields

3d∑

|j1|�L/4

∫

|x1|>L/2f (1)(x1 − j1) dx1

� 3dn0|| sup|j1|�L/4

∫

|x1|>L/2f (1)(x1 − j1) dx1 (59)

as an upper bound for the remainder for all L � 8. Here the inequality followsfrom the estimate #{|j1| � L/2} � n0|| for some n0 < ∞ and all L > 1. Theproof is completed by employing a result for f (1) analogous to (28). �

4.3.2. Proof of Theorem 2.8 – First Part: Quantum-Classical Regime

We fix r0 > 1/(2µ(1)(0)) large enough to ensure the validity of (49) in Proposi-tion 4.6. For a given energy E > 0 we then pick

L :=(

c2nu

2r30E

)1/2

(60)

where the constants c2 and nu have been fixed in Lemma 4.7 and Remark 4.8.Finally, we choose the cuboid from (45) and set R := (r0L)2/α2(1−γ ). Proposi-tion 4.6 and (55) then yield the estimate

P{ω ∈ � : λ0(Hχ

(Vω)) < E}� P

{ω ∈ � : 1

||∑

j∈Zd∩

µ(1)ω (j ) <

(r0L)2

c2nu

(2E + c2L−α1(1−γ ))

}


� P

{ω ∈ � : 1

||∑

j∈Zd∩

µ(1)ω (j ) <

2

r0

}(61)

provided E > 0 is small enough, equivalently L is large enough. Here the lastinequality results from (60) and from the first inequality in (qm/cl), which impliesthat c3r

30L2 � c2nuL

α1(1−γ ) for large enough L > 0. Since 2/r0 � µ(1)(0) byassumption on r0, the right-hand side of (61) is the probability of a large-deviationevent [4, 7]. Consequently, there exists some constant c4 > 0 (which is independentof L) such that (61) is estimated from above by

exp[−c4||] � exp[−c4nuLd1(r0L)2γ2/(1−γ )]

= exp[−c5E−d1/2−γ2/(1−γ )]. (62)

Here the existence of a constant c5 > 0 ensuring the validity of the last equalityfollows from (60). Inserting this estimate in the right-hand side of (23) completesthe first part of the proof of Theorem 2.8 for the quantum-classical regime, sincethe pre-factor in the upper bound in Proposition 3.2 is negligible.

4.4. CLASSICAL REGIME

Throughout this subsection we suppose that (cl) holds. For an asymptotic evalua-tion of the upper bound in Proposition 3.2 in the present case, we define

βk := 2

dk

γk

1 − γ= 2

αk(1 − γ ), k ∈ {1, 2} (63)

and construct a lower bound on the lowest Mezincescu eigenvalue λ0(Hχ

int0

(Vω))

showing up in the right-hand side of (23) when choosing = int0 the open unit

cube there.


For every L > 1 and ω ∈ � the potential Vω,L: Rd → [0, ∞[ given by

Vω,L(x) :=∫

|y1|>Lβ1

|y2|>Lβ2

f (x − y)µ(1)ω (dy) (64)

in terms of the regularised Borel measure µ(1)ω , provides a lower bound on Vω.

Therefore λ0(Hχ

int0

(Vω)) � λ0(Hχ

int0

(Vω,L)). It will be useful to collect some facts

related to Vω,L.

LEMMA 4.9. Let L > 1 and define VL: Rd → [0, ∞[ by

VL(x) :=∑

|j1|>Lβ1 −1|j2|>Lβ2 −1

supy∈j

f (x − y). (65)


Then we have Vω,L � VL for every ω ∈ �. Moreover, the supremum supx∈0VL(x)

is arbitrarily small for large enough L > 1.Proof. The first assertion follows analogously as in Lemma 4.5. The second one

derives from the second inequality in (12). �Remark 4.10. It is actually not difficult to prove that there exists some constant

0 < C < ∞ (which is independent of L) such that supx∈0VL(x) � CL−2 for

large enough L > 0.

The next proposition contains the key estimate on the lowest Mezincescu eigen-value in the classical regime. In contrast to the quantum-classical regime, the spe-cific choice of the cut-off made in (64) is irrelevant as far as the applicability ofTemple’s inequality in the subsequent proposition is concerned. The chosen lengthscales Lβ1 and Lβ2 will rather become important later on.

PROPOSITION 4.11. Let int0 be the open unit cube. Then the lowest eigenvalue

of Hχ

int0

(Vω,L) is bounded from below according to

λ0(Hχ

int0

(Vω,L)) � 1

2

∫

0

Vω,L(x)ψ(x)2 dx (66)

for all ω ∈ � and large enough L > 1. [Recall the definition of ψ at the beginningof Subsection 3.1.]

Proof. The proof again parallels that of Proposition 4.3. In a slight abuse ofnotation, let ψ denote the restriction of ψ to int

0 throughout this proof. Temple’sinequality [35, Theorem XIII.5] together with the fact that H

χ

int0

(0)ψ = 0 yields

the lower bound

λ0(Hχ

int0

(Vω,L)) � 〈ψ, Vω,Lψ〉 − 〈Vω,Lψ, Vω,Lψ〉λ1(H

χ

int0

(0)) − 〈ψ, Vω,Lψ〉 (67)

provided that the denominator is strictly positive. To check this we employ Lem-ma 4.9 and take L > 1 large enough such that 〈ψ, Vω,Lψ〉 � λ1(H

χ

int0

(0))/2. (Note

that λ1(Hχ

int0

(0)) is independent of L.) To estimate the numerator in (67) from

above, we use the bound 〈Vω,Lψ, Vω,Lψ〉 � 〈ψ, Vω,Lψ〉 supx∈0VL(x). Together

with Lemma 4.9 this yields 〈Vω,Lψ, Vω,Lψ〉 � 〈ψ, Vω,Lψ〉λ1(Hχ

int0

(0))/4 for large

enough L > 1. �Remark 4.12. The simple lower bound λ0(H

χ

int0

(Vω,L)) � infx∈0 Vω,L(x),

which was employed in [21], would yield a result similar to (72) below, but atthe price of assuming that the lower bound in (12) holds pointwise.


We proceed by constructing a lower bound on the right-hand side of (66). Forthis purpose we set

:=⋃

2Lβ1 <|j1|�4Lβ1

2Lβ2 <|j2|�4Lβ2

j (68)

an annulus-shaped region.

LEMMA 4.13. There exists a constant c6 > 0 (which is independent of ω and L)such that

∫

0

Vω,L(x)ψ(x)2 dx � c6

L2/(1−γ )µ(1)

ω () (69)

for large enough L > 0.Proof. Pulling out the strictly positive infimum of ψ2, using Fubini’s theorem

and omitting a positive term, we estimate∫

0

Vω,L(x)ψ(x)2 dx � infz∈0

ψ(z)2∫

(∫

0

f (x − y) dx

)µ(1)

ω (dy). (70)

Assumption 2.4 implies that the estimate∫0

f (x − y) dx � fu/[(3Lβ1)α1 +(3Lβ2)α2] holds for all y ∈ and large enough L > 1. This completes the proof,since αkβk = 2/(1 − γ ) for both k ∈ {1, 2}. �

Remark 4.14. There exists some constant nu > 0 such that the number of lat-tice points in can be bounded from below according to || � nuL

β1d1+β2d2 =nuL

2γ /(1−γ ) for all L > 1. Lemma 4.13 thus implies the inequality∫

0

Vω,L(x)ψ(x)2 dx � c6nu

L2||−1µ(1)

ω () (71)

for large enough L > 1.

4.4.2. Proof of Theorem 2.8 – First Part: Classical Regime

For a given energy E > 0 we let L := (c6nuµ(1)(0)/4E)1/2, where the constant

c6 and nu have been fixed in Lemma 4.13 and Remark 4.14. Proposition 4.11 andEquation (71) then yield the estimate

P{ω ∈ � : λ0(Hχ

int0

(Vω)) < E}

� P

{ω ∈ � : 1

||∑

j∈Zd∩

µ(1)ω (j ) <

2EL2

c6nu

}(72)


provided E > 0 is small enough, equivalently L is large enough. Since 2EL2/

c6nu = E[µ(1)ω (0)]/2 and the random variables are independent and identically

distributed, the last probability is that of a large deviation event [4, 7]. Conse-quently, there exists some c7 > 0 such that the right-hand side of (72) is boundedfrom above by

exp[−c7||] � exp[−c7nuL2γ /(1−γ )]

= exp

[−c7nu

(c6nuµ

(1)(0)

4E

)γ /(1−γ )]. (73)

Since the pre-factor in the upper bound in Proposition 3.2 is negligible, insert-ing (72) together with (73) in the right-hand side of (23) completes the first part ofthe proof of Theorem 2.8 for the classical regime.

5. Lower Bound

To complete the proof of Theorem 2.8, it remains to asymptotically evaluate thelower bound in Proposition 3.2 for small energies. This is the topic of the presentsection. In order to do so, we first construct an upper bound on the lowest Dirichleteigenvalue showing up in the left-hand side of (23) when choosing

:=⋃

|j |<L/4

j

int

(74)

with L > 0 there. By construction is open and compatible with the lattice.

5.1. UPPER BOUND ON LOWEST DIRICHLET EIGENVALUE

The following lemma basically repeats [21, Prop. 5] and its corollary.

LEMMA 5.1. Let denote the open cube (74). There exist two constant 0 <

C1, C2 < ∞ (which are independent of ω and L) such that

λ0(HD (Vω)) � C1||−1

∫

Vω(x) dx + C2L−2 (75)

for all ω ∈ � and all L > 1.Proof. We let θ ∈ C∞

c (0) denote a smoothed indicator function of the cube{x ∈ R

d : |x| < 1/4} ⊂ 0 and set θL(x) := θ(x/||1/d) for all x ∈ .Choosing the product of θL ∈ C∞

c () and the ground-state function ψ of H(0)

as the variational function in the Rayleigh–Ritz principle we obtain

λ0(HD (Vω))〈θLψ, θLψ〉 � 〈θLψ, HD

(Vω)θLψ〉= 〈θLψ, VωθLψ〉 + 〈(∇θL)ψ, (∇θL)ψ〉� sup

y∈0

ψ(y)2

[∫

Vω(x) dx + ||1−2/d

∫

0

|∇θ(x)|2 dx

]. (76)


Here the equality uses Hχ

(0)ψ = 0 and integration by parts. Observing that〈θLψ, θLψ〉 � 2−d || infx∈0 ψ(x)2 and that there is some constant C > 0 suchthat ||1/d � CL for all L > 1, completes the proof. �

Our next task is to bound the integral in the right-hand side of (75) from above.For this purpose it will be useful to introduce the cuboid

:=⋃

|j1|�2Lβ1

|j2|�2Lβ2

j, (77)

which contains the cube defined in (74). Here and in the following we usethe abbreviation βk := max{1, 2/αk(1 − γ )} = 2/dk max{dk/2, γk/(1 − γ )}, fork ∈ {1, 2}.

LEMMA 5.2. Let L > 0 and define the random variable

Wω(L) := ||−1∫

Rd\

(∫

f (x − y) dx

)µω(dy). (78)

Then the following three assertions hold true:

(i) ||−1∫

Vω(x) dx � ‖f ‖1µω() + Wω(L) for all ω ∈ � and all L > 0.(ii) There exists some constant 0 < C3 < ∞ (which is independent of ω and L)

such that

P{ω ∈ � : Wω(L) � C3L−2} � 1

2 (79)

for large enough L.(iii) The random variables µ() and W(L) are independent for all L > 0.

Proof. For a proof of the first assertion we decompose the domain of integrationand use Fubini’s theorem to obtain

∫

Vω(x) dx =∫

(∫

f (x − y) dx

)µω(dy)

+∫

Rd\

(∫

f (x − y) dx

)µω(dy)

� ‖f ‖1µω() + ||Wω(L). (80)

Here the inequality results from the estimate∫

f (x − y) dx �∫

Rd f (x) dx =:‖f ‖1 valid for all y ∈ R

d . This yields Lemma 5.2(i) since 1 � ||. For a proof ofthe second assertion, we employ Chebychev’s inequality

P{ω ∈ � : Wω(L) � C3L−2}

� L2

C3||−1

E

[∫

Rd\

(∫

f (x − y) dx

)µ(dy)

]


= L2

C3||−1

∫

(∫

Rd\f (x − y) dx

)µ(dy)

� L2

C3µ(0) sup

y∈

∫

Rd\f (x − y) dx. (81)

Here the inequality uses the fact that the intensity measure µ is Zd-periodic. The

inner integral is in turn estimated from above in term of two integrals involving themarginal impurity potentials f (1) and f (2) (recall the definitions (24) and (25))

∫

Rd\f (x − y) dx �

∫

|x1|>Lβ1

f (1)(x1 − y1) dx1

+∫

|x2|>Lβ2

f (2)(x2 − y2) dx2

� CL−2. (82)

Here the existence of some 0 < C < ∞ ensuring the last inequality for all |y| �L/2 (that is in particular; for all y ∈ ) and sufficiently large L � 4 followsfrom (28) and the fact that βkαk(1 − γ ) � 2. Taking C3 in (81) large enough yieldsthe second assertion. The third assertion is a consequence of Assumption 2.1(ii). �

5.2. PROOF OF THEOREM 2.8 – FINAL PARTS

For a given energy E > 0 we choose

L :=(

3 max{C2, C3}E

)1/2

, (83)

where the constants C2 and C3 were fixed in Lemmas 5.1 and 5.2, respectively.Moreover, we pick the cube from (74) and the cuboid from (77). EmployingLemmas 5.1 and 5.2 we estimate the probability in the right-hand side of (23)according to

P{ω ∈ � : λ0(HD (Vω)) < E}

� P({ω ∈ � : λ0(HD (Vω)) < E} ∩ {ω ∈ � : Wω(L) < C3L

−2})� P

({ω ∈ � : µω() <

max{C2, C3}L−2

C1‖f ‖1

}

∩ {ω ∈ � : Wω(L) < C3L−2}

). (84)

Since the random variables µ() and W(L) are independent, the probability in (84)factorises. Thanks to (79) the probability of the second event is bounded frombelow by 1/2 provided that L is large enough, equivalently, that E > 0 is small


enough. Employing the decomposition (74) of into || unit cubes of the lat-tice Z

d , we have µω() = ∑j∈∩Zd µω(j ) such that the probability of the first

event in (84) is bounded from below by

P

{ω ∈ � : µω(j) <

max{C2, C3}L−2

C1‖f ‖1|| for all j ∈ ∩ Zd

}. (85)

By construction of there is some constant n0 > 0 such that || � n0Lβ1d1+β2d2 .

Abbreviating C4 := max{C2, C3}/(C1‖f ‖1n0) and ϑ := 2 + β1d1 + β2d2, andusing the fact that the random variables µ(j) are independent and identicallydistributed (by virtue of Assumption 2.1), the last expression (85) may be boundedfrom below by

P{ω ∈ � : µω(0) < C4L−ϑ}n0L

β1d1+β2d2

� (C4L−ϑ)κn0L

β1d1+β2d2

= exp[C5(log Eϑ/2 + log C6)E−(β1d1+β2d2)/2]. (86)

Here the first inequality derives from Assumption 2.1 on the probability measureof µ(0). Moreover, the existence of two constants 0 < C5, C6 < ∞ ensur-ing the validity of the equality follows from (83). Since the choice (83) of theenergy-dependence of L guarantees that the pre-factor in the lower bound in Propo-sition 3.2 is negligible, the proof of Theorem 2.8 is completed by inserting (86) inthe left-hand side of (23).

Appendix A. Proof of Mixing of Random Borel Measure

The purpose of this short appendix is to prove Lemma 2.3. We let (n) := ⋃|j |�n j

with n ∈ N. Moreover, let M((n)) ⊂ M(Rd) denote the set of Borel measureswith support in (n) and let B(Mn) be the smallest σ -algebra, which renders themappings M((n)) � ν �→ ν() measurable for all Borel sets ⊂ (n). Theirunion R := ⋃

n∈NB(Mn) satisfies:

(i) R generates the σ -algebra B(M).(ii) R is a semiring.

The first assertion holds by definition of B(M). To check the second one we notethat ∅ ∈ R. Moreover, for every M , M ′ ∈ R there exists some n ∈ N such that

M, M ′ ∈ B(Mn) (87)

and hence M ∩ M ′ ∈ B(Mn) ⊂ R and M\M ′ ∈ B(Mn) ⊂ R.Our next aim is to prove the claimed limit relation (11) for all M , M ′ ∈ B(Mn)

with n ∈ N arbitrary. Assumption 2.1(ii) ensures that the events TjM ⊂M((n) + j) and M ′ ⊂ M((n)) are stochastically independent for all j ∈ Z

d

with ((n) + j) ∩ (n) = ∅, such that

P {TjM ∩ M ′} = P {TjM}P {M ′} = P {M}P {M ′}. (88)

Here the last equality is a consequence of Assumption 2.1(i).


Thanks to (87) we have thus proven the validity of (88) for all M , M ′ ∈ R.Lemma 2.3 now follows from [6, Lemma 10.3.II], which is a monotone-classargument.

Remark A.1. We proved above that the random potential Vω is mixing under ourassumptions. Note, that mixing is actually a property of the probability measure Pwith respect to the shifts {Tj }. However, the potential Vω will not satisfy strongermixing condition such as φ-mixing. In fact, as a rule, the potential may even bedeterministic (in the technical sense of this notion, see, e.g., [17]), which allowsmixing, but not φ-mixing. For further references to this see [1, 19].

Acknowledgement

We are grateful to Hajo Leschke for helpful remarks. This work was partiallysupported by the DFG within the SFB TR 12.

References

1. Billingsley, P.: Convergence of Probability Measures, Wiley, 1968.2. Broderix, K., Hundertmark, D., Kirsch, W. and Leschke, H.: The fate of Lifshits tails in

magnetic fields, J. Statist. Phys. 80 (1995), 1–22.3. Carmona, R. and Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Birkhäuser,

Boston, 1990.4. Durrett, R.: Probability: Theory and Examples, Duxbury, Belmont, 1996.5. Donsker, M. D. and Varadhan, S. R. S.: Asymptotics of the Wiener sausage, Comm. Pure Appl.

Math. 28 (1975), 525–565. Errata: ibid, p. 677.6. Daley, D. J. and Vere-Jones, D.: An Introduction to the Theory of Point Processes, Springer,

New York, 1988.7. Dembo, A. and Zeitouni, O.: Large Deviations Techniques and Applications, Springer, New

York, 1998.8. Erdos, L.: Lifschitz tail in a magnetic field: The nonclassical regime, Probab. Theory Related

Fields 112 (1998), 321–371.9. Erdos, L.: Lifschitz tail in a magnetic field: Coexistence of the classical and quantum behavior

in the borderline case, Probab. Theory Related Fields 121 (2001), 219–236.10. Hundertmark, D., Kirsch, W. and Warzel, S.: Lifshits tails in three space dimensions: Impurity

potentials with slow anisotropic decay, Markov Process. Related Fields 9 (2003), 651–660.11. Hupfer, T., Leschke, H. and Warzel, S.: Poissonian obstacles with Gaussian walls discriminate

between classical and quantum Lifshits tailing in magnetic fields, J. Statist. Phys. 97 (1999),725–750.

12. Hupfer, T., Leschke, H. and Warzel, S.: The multiformity of Lifshits tails caused by randomLandau Hamiltonians with repulsive impurity potentials of different decay at infinity, AMS/IPStud. Adv. Math. 16 (2000), 233–247.

13. Kallenberg, O.: Random Measures, Akademie-Verlag, Berlin, 1983.14. Kirsch, W.: Random Schrödinger operators: A course, In: H. Holden and A. Jensen (eds),

Schrödinger Operators, Lecture Notes in Phys. 345, Springer, Berlin, 1989, pp. 264–370.15. Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators,

Duke Math. J. 98 (1999), 335–369. Erratum: mp_arc 00-389.


16. Klopp, F.: Une remarque á propos des asymptotiques de Lifshitz internes, C.R. Acad. Sci. ParisSer. I 335 (2002), 87–92.

17. Kirsch, W., Kotani, S. and Simon, B.: Absence of absolutely continuous spectrum for someone-dimensional random but deterministic Schrödinger operators, Ann. Inst. H. Poincare Phys.Théor. 42 (1985), 383–406.

18. Kirsch, W. and Martinelli, F.: On the ergodic properties of the spectrum of general randomoperators, J. Reine Angew. Math. 334 (1982), 141–156.

19. Kirsch, W. and Martinelli, F.: Large deviations and Lifshitz singularity of the integrated densityof states of random Hamiltonians, Comm. Math. Phys. 89 (1983), 27–40.

20. Kirsch, W. and Martinelli, F.: On the essential self adjointness of stochastic Schrödingeroperators, Duke Math. J. 50 (1983), 1255–1260.

21. Kirsch, W. and Simon, B.: Lifshits tails for periodic plus random potentials, J. Statist. Phys. 42(1986), 799–808.

22. Kirsch, W. and Simon, B.: Comparison theorems for the gap of Schrödinger operators, J. Funct.Anal. 75 (1987), 396–410.

23. Klopp, F. and Wolff, T.: Lifshitz tails for 2-dimensional random Schrödinger operators, J. Anal.Math. 88 (2002), 63–147.

24. Lang, R.: Spectral Theory of Random Schrödinger Operators, Lecture Notes in Math. 1498,Springer, Berlin, 1991.

25. Leschke, H., Müller, P. and Warzel, S.: A survey of rigorous results on random Schrödingeroperators for amorphous solids, Markov Process. Related Fields 9 (2003), 729–760.

26. Leschke, H. and Warzel, S.: Quantum-classical transitions in Lifshits tails with magnetic fields,Phys. Rev. Lett. 92 (2004), 086402 (1–4).

27. Lifshitz, I. M.: Structure of the energy spectrum of the impurity bands in disordered solidsolutions, Soviet Phys. JETP 17 (1963), 1159–1170. Russian original: Zh. Eksper. Teoret. Fiz.44 (1963), 1723–1741.

28. Mezincescu, G. A.: Internal Lifshitz singularities for disordered finite-difference operators,Comm. Math. Phys. 103 (1986), 167–176.

29. Mezincescu, G. A.: Lifschitz singularities for periodic operators plus random potential,J. Statist. Phys. 49 (1987), 1181–1190.

30. Mezincescu, G. A.: Internal Lifshitz singularities for one-dimensional Schrödinger operators,Comm. Math. Phys. 158 (1993), 315–325.

31. Mine, T.: The uniqueness of the integrated density of states for the Schrödinger operators forthe Robin boundary conditions, Publ. Res. Inst. Math. Sci., Kyoto Univ. 38 (2002), 355–385.

32. Nakao, S.: On the spectral distribution of the Schrödinger operator with random potential,Japan. J. Math. 3 (1977), 111–139.

33. Pastur, L. A.: Behavior of some Wiener integrals as t → ∞ and the density of statesof Schrödinger equations with random potential, Theoret. Math. Phys. 32 (1977), 615–620.Russian original: Teoret. Mat. Fiz. 6 (1977), 88–95.

34. Pastur, L. and Figotin, A.: Spectra of Random and Almost-Periodic Operators, Springer, Berlin,1992.

35. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators,Academic, New York, 1978.

36. Simon, B.: Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526. Erratum:Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526.

37. Simon, B.: Lifshitz tails for the Anderson model, J. Statist. Phys. 38 (1985), 65–76.38. Simon, B.: Internal Lifshitz tails, J. Statist. Phys. 46 (1987), 911–918.39. Stoyan, D., Kendal, W. S. and Mecke, J.: Stochastic Geometry and Its Applications, Wiley,

Chichester, 1987.40. Stollmann, P.: Lifshitz asymptotics via linear coupling of disorder, Math. Phys. Anal. Geom. 2

(1999), 2679–2689.


41. Stollmann, P.: Caught by Disorder: Bound States in Random Media, Birkhäuser, Boston, 2001.42. Veselic, I.: Integrated density of states and Wegner estimates for random Schrödinger operators,

Contemp. Math. 340 (2004), 97–183.43. Warzel, S.: On Lifshits Tails in Magnetic Fields, Logos, Berlin, 2001. PhD thesis, University

Erlangen–Nürnberg (2001).


Pair Correlation Statistics for the Zerosof Lamé Polynomials

ALAIN BOURGETDepartment of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton,Ontario L8S 4K1, Canada. e-mail: [email protected]

(Received: 24 July 2003; in final form: 26 May 2004)

Abstract. The joint eigenfunctions of a quantum completely integrable system can naturally bedescribed in terms of products of Lamé polynomials. In this paper, we compute the limiting paircorrelation distribution for the zeros of Lamé polynomials in various thermodynamic, asymptoticregimes. We give results both in the mean and pointwise, for an asymptotically full set of values ofthe parameters α0, . . . , αN .

Mathematics Subject Classifications (2000): 81R12, 53A55.

Key words: pair correlation, quantum integrable system, Lamé polynomials.

1. Introduction

In a recent paper [BT], it was shown that the limiting level-spacings distributionsfor the zeros of Lamé polynomials follow a Poisson distribution. In this paper, westudy another important statistic associated to quantum integrable systems, namelythe pair correlation distribution (PCD) of the zeros of the joint eigenfunctions.

We briefly recall the setting. For any given set of N + 1 parameters 0 < α0 <

· · · < αN , let Sij

k be the kth elementary symmetric polynomial in α0, . . . , αN withαi and αj deleted. The partial differential operators

Pk :=∑

i<j

Sij

k (α0, . . . , αN)

(xi

∂

∂xj

− xj

∂

∂xi

)2

, k = 0, . . . , N − 1,

acting on C∞(SN) form a quantum completely integrable system (QCI). Indeed,one can easily check that

(i) P0 = �SN ,(ii) [Pi, Pj ] = 0 for all i, j = 0, . . . , N − 1.

This QCI naturally depends on the parameters α0, . . . , αN , and thus constitutes anensemble called the Lamé ensemble [BT].

Since the Pj ’s are jointly elliptic, they possess a Hilbert basis of joint eigen-functions of spherical harmonics, the so-called generalized Lamé harmonics [Vo].

ALAIN BOURGET

Using elliptic-spherical coordinates (u1, . . . , uN) ∈ (α0, α1)×· · ·× (αN−1, αN) toseparate variables [BT, T], we can write the joint eigenfunctions in the form

ψ(u1, . . . , uN) =N∏

j=1

Sm(√

uj − α0, . . . ,√

uj − αN)φ(uj ).

Here, φ is a polynomial, and Sm(x0, . . . , xN), m = 0, . . . , N + 1, denotes the mthelementary symmetric function on N + 1 variables. Furthermore, the functionψ(x) = Sm(

√x − α0, . . . ,

√x − αN)φ(x) is a solution of the Lamé differential

equation

N∏

ν=0

(x − αν)d2ψ

dx2(x) + 1

2

N∑

ν=0

∏

µ �=ν

(x − αµ)dψ

dx(x) + V (x)ψ(x) = 0, (1)

where V (x) is a polynomial of degree N − 1 depending linearly on the jointeigenvalues (λ0, . . . , λN−1) ∈ Spec(P0, . . . , PN−1). In the special case m = 0,the solutions ψ(x) are called Lamé polynomials.

Accordingly, we denote the Lamé polynomials of degree K by φ(K)

1 (x), . . . ,

φ(K)

σ(N,K)(x), where σ(N, K) is the dimension of spherical harmonics of degree K

on SN . We also denote the zeros of φ

(K)j (x) by θ

(j)

1 (α) < · · · < θ(j)

K (α). In our mainresults, we compute the asymptotic weak-limit for the pair correlation distributionof the zeros averaged over the set of Kth degree Lamé polynomials. More precisely,we consider

dρAvPC(x;N, K, α)

:= 1

σ(N, K)

σ(N,K)∑

l=1

1

K

K∑

i �=j

δ(x − K(θ(l)j (α) − θ

(l)i (α))), (2)

where α ∈ �N , the positive Weyl chamber defined by

�N := {α = (α0, . . . , αN) ∈ [0, 1]N+1 : α0 < · · · < αN }.We henceforth put normalized Lebesgue measure d−α := (N + 1)!dα on �N ,

and denote the expectation with respect to the probability measure d−α by E�N .Our first two results show that the averaged spacings between zeros of Lamé

polynomials have a Poissonian distribution. In other words, the averaged zerosdistribute like those between members of a sequence of random numbers in theunit interval.

THEOREM 1.1. Assume that K = o(N1/2) as N → ∞. Then, we have

w-limK→∞

E�N [dρAvPC(x;N, K, α)] = dx.

In other words,

PAIR CORRELATION STATISTICS

limK→∞

∫

�N

dρAvPC(φ;N, K, α) d−α =

∫ b

a

φ(x) dx

for any continuous function φ supported in (a, b), where −∞ < a < b < ∞.

By computing the variance of the pair correlation measures dρAvPC , we obtain the

following pointwise version of Theorem 1.1.

THEOREM 1.2. Assume that K = o(N1/2) as N → ∞. Then, there exists asubset JN ⊆ �N of measure asymptotically one such that for any α ∈ JN ,

limK→∞ dρAv

PC(φ;N, K, α) =∫ b

a

φ(x) dx

for any continuous function φ supported in (a, b), where −∞ < a < b < ∞.

The main idea of the proofs of Theorems 1.1 and 1.2 is based on a simple conse-quence of the Heine–Stieltjes theorem [Sz]. In Section 2, we recall this importantresult. We then present in Sections 3 and 4 the proofs of Theorems 1.1 and 1.2.Finally, as a consequence of Theorem 1.2, we compute, in the last section of thepaper, the averaged pair correlation function of the zeros of Lamé polynomials.

Remarks. (i) Clearly, both Theorems 1.1 and 1.2 are still valid for the zeros ofjoint eigenfunctions of the form ψ(x) = Sm(

√x − α0, . . . ,

√x − αN)φ(x) with

m �= 0. Indeed, for each fixed m �= 0, the zeros of ψ(x) are exactly the zeros of theassociated polynomials φ(x).

(ii) As in [BT], there are two asymptotic parameters that enter into our analysis:N , the dimension of the base space S

N , and K , the degree of the joint eigenfunc-tions. Our results are then concerned with the limiting PCD of zeros of the jointeigenfunctions of the operators Pj ’s on spheres of increasing dimension wherewe assume that K satisfies K(N) � N1/2 as N → ∞. This asymptotic regimecan then be viewed as a thermodynamical limit (N → ∞). Of course, it wouldbe of interest to determine the limiting PCD in other asymptotic regimes wherewe allow K to grow faster than N1/2. In particular, it would be of great interest tocompute the PCD in the purely semiclassical regime where N is fixed and K → ∞.

(iii) As we mentioned above, similar techniques were applied to compute theaveraged level spacings distribution of the zeros of Lamé polynomials [BT]. Ac-tually, we showed that the limiting averaged level spacings follow an exponentialdistribution. These techniques can also be applied to show that the zeros of any VanVleck polynomial follow a uniform distribution as N → ∞ (see [B]).

(iv) In a recent paper, Martinez and Saff [MS] have also obtained asymptoticresults for the zeros of Van Vleck and Lamé polynomials. However, their resultsdiffer from ours in the sense that the limit distribution is described in terms of anequilibrium measure with external field, generated by charges at the singular pointsα0, . . . , αN of the generalized Lamé equation.

ALAIN BOURGET

(v) The zeros of Lamé polynomials can naturally be interpreted as equilibriumpositions of an electrostatic system with logarithmic potential [G, I]. However, inthis paper we do not use this extra information, our arguments are purely combina-torial and based on the Heine–Stieltjes theorem. It would be, of course, very inter-esting to see if one can use this fact to improve the results we present in this paper.

2. The Heine–Stieltjes Theorem

One of the key idea in the proof of Theorem 1.1 relies on the fact that one canuniquely characterize each Lamé polynomials by its zeros distribution on the in-tervals (α0, α1), . . . , (αN−1, αN). This important characterization is based on thefollowing result (see [Sz]).

THEOREM 2.1 (Heine–Stieltjes). Let A(x) be the polynomial of degree N + 1given by

A(x) = (x − α0) · (x − α1) · · · (x − αN),

where 0 < α0 < α1 < · · · < αN , and B(x) be a polynomial of degree N satisfyingthe condition

B(x)

A(x)= ρ0

x − α0+ · · · + ρN

x − αN

,

for given numbers ρν > 0 (ν = 0, . . . , N ). Then, there are exactly

σ(N, K) = (N + K − 1)!K!(N − 1)!

polynomials C(x) of degree N − 1 for which the differential equation

A(x)d2φ

dx2+ 2B(x)

dφ

dx+ C(x)φ = 0 (3)

has a polynomial solution of degree K > 0. In addition, for each of the σ(N, K)

solutions, φ(x), the zeros are simple and uniquely determined by their distributionin the intervals (α0, α1), . . . , (αN−1, αN).

In other words, given any N nonnegative integers m1, . . . , mN such that m1 +· · · + mN = K , there exists a unique (monic) Lamé polynomial φ(x) of degree K

having mj zeros lying inside the interval (αj−1, αj ) for j = 1, . . . , N .Consequently, we denote the zeros of φ(x) by θ1(α; l) � · · · � θK(α; l), where

α := (α0, . . . , αN), whereas l = (l1, . . . , lK), 1 � l1 � · · · � lK � N , denote theconfiguration of the zeros. That is, θ1(α; l) is the smallest zero lying in the interval(αl1−1, αl1), the next zero θ2(α; l) is contained in the interval (αl2−1, αl2) and so on.


3. Proof of Theorem 1.1

By a simple density argument, we can assume without loss of generality that φ ∈C1

0((a, b)). Following the notation introduced in Section 2, we can rewrite dρAvPC in

the more convenient form

dρAvPC(φ;K, N, α) = 1

σ(N, K)

∑

|l|=K

1

K

K∑

i �=j

φK(θj (α; l) − θi(α; l)), (4)

where |l| = K denotes all K-tuples (l1, . . . , lK) of positive integers satisfying1 � l1 � · · · � lK � N . Here, we used dρAv

PC to denote dρAvPC(φ;K, N, α), and

φK(x) to denote φ(Kx). We will use these shorter notations throughout the rest ofthe text.

The first step of the proof consists of replacing each of the zeros θj (α; l) by αlj

in dρAvPC . Recall, the zeros of any Lamé polynomials satisfy the inequalities

αlj −1 < θj(α; l) < αlj , for all j = 1, . . . , K. (5)

The set of inequalities in (5) naturally suggests to expand each of the expressionsθj (α; l) − θi(α; l) in dρAv

PC in a first-order Taylor series around αlj − αli . We thenobtain

E�N [dρAvPC] = 1

σ(N, K)

∑

|l|=K

1

K

K∑

i �=j

∫

�N

φK(αlj − αli ) d−α + R1(φ, N, K),

where the error term, R1(φ, N, K), is given by

R1(φ, N, K) = 1

σ(N, K)

∑

|l|=K

K∑

i �=j

∫

�N

φ′K(ξij (α))[(θj (α; l) − αlj )

− (θi(α; l) − αli )] d−α,

for some ξij (α) ∈ (0, 1).The fact that R1(φ, N, K) � K2/N can be obtained using Lemma 3.1 in [BT]:

∫

�N

|θ(α; l) − αlj | d−α � 2

N + 2for all j = 1, . . . , K,

and the simple computation:

|R1(φ, N, K)| � 1

σ(N, K)

∑

|l|=K

∑

i �=j

2

N + 2‖φ‖C1

= O

(K2

N

).

Summing up, we have shown that

E�N [dρAvPC] = 1

σ(N, K)

∑

|l|=K

1

K

∑

i �=j

∫

�N

φK(αlj − αli ) d−α + O

(K2

N

). (6)

ALAIN BOURGET

Under the assumption K = o(N1/2), we can restrict ourselves to configurationsthat have at most one zero inside each intervals (αj , αj+1). Indeed, the number ofconfigurations of the former type is obviously equal to

(N

K

), since it corresponds to

the number of ways one can choose K intervals (αl1, αl1+1), . . . , (αlK , αlK+1), eachcontaining exactly one zero, among a total of N intervals. Moreover, we also have

(N

K

)

σ(N, K)= N !

K!(N − K)!K!(N − 1)!

(N + K − 1)!= N(N − 1) · · · (N − K + 1)

(N + K − 1)(N + K − 2) · · · N

=K−1∏

j=1

1

1 + K−1N−j

=K−1∏

j=1

(1 + O

(K

N

))

= 1 + O

(K2

N

). (7)

Consequently, we can replace the sum over all configurations l in (6) by thesimpler sum over the configurations l that have at most one zero in each of thesubintervals (αj , αj+1). This yields the following result.

PROPOSITION 3.1. The estimate (6) for E�N [dρAvPC] simplify to

E�N [dρAvPC] = 1

(N

K

)∑

|l|=K

1

K

∑

i �=j

∫

�N

φK(αlj − αli ) d−α + O

(K2

N

), (8)

where |l| = K denotes all K-tuples (l1, . . . , lK) of positive integers satisfying1 � l1 < · · · < lK � N .

Proof. We decompose the sum over all the configurations l in (6) into twodisjoint sums. In the first one, we sum over the configurations l having at mosta single zero in each interval (α0, α1), . . . , (αN−1, αN). The second sum is takenover configurations l′ having at least one interval (αj , αj+1) containing more thanone zero. More precisely, we can rewrite (6) as

E�N [dρAvPC] = 1

σ(N, K)

[ ∑

|l|=K

+∑

|l′|=K

][1

K

∑

i �=j

∫

�N

φK(αlj − αli ) d−α]

+ O

(K2

N

),

where |l′| = K denotes all K-tuple (l1, . . . , lK) of positive integers satisfying 1 �l1 � · · · � lK � N with lj = lj+1 for at least one j ∈ {1, . . . , K}. Clearly, the setof all configurations l is the disjoint union of the configurations l and l′.


The rest of the proof is then an immediate consequence of (7) and the followingupper bound for 1

K

∑Ki �=j

∫�N φK(αlj − αli ) d−α, i.e.

∣∣∣∣∣1

K

K∑

i �=j

∫

�N

φK(αlj − αli ) d−α

∣∣∣∣∣ � ‖φ‖L1 . (9)

To prove (9), we use the invariance of the function∑

i �=j φK(αj − αi) underthe permutations in SN+1, the symmetric group of N + 1 elements, to replace theintegration over �N by the simpler integration over the unit cube [0, 1]N+1. As aconsequence, we obtain

1

K

K∑

i �=j

∫

�N

φK(αlj − αli ) d−α = 1

K

K∑

i �=j

∫ 1

0

∫ 1

0φK(αlj − αli ) dαlj dαli

= K

∫ 1

0

∫ 1

0φK(y − x) dy dx. (10)

The last equality follows from the basic fact that variables of integration are dummyvariables. To further simplify the integral on the left-hand side of (10), we makethe change of variables v = Ky and u = K(y − x). Moreover, we use the fact thatφ is supported in the finite interval (a, b) to finally obtain

K

∫ 1

0

∫ 1

0φK(y − x) dy dx

= 1

K

[ ∫ 0

−K

∫ K+u

0φ(x) dv du +

∫ K

0

∫ K

u

φ(u) dv du

]

= 1

K

∫ K

−K

φ(x)(K − |u|) du

�∫ b

a

|φ(u)| du

= ‖φ‖L1 .

This completes the proof of (9) and the proof of Proposition 3.1. �For each (i, j) ∈ {1, . . . , N}2, the occurrence of φK(αj − αi) in (8) is

(N−2K−2

)

since there are(N−2K−2

)ways of distributing the remaining K −2 zeros in the remain-

ing N −2 intervals while keeping the other two zeros fixed (one zero in the interval(αi, αi+1) and one zero in the interval (αj , αj+1)). Thus, we have

E�N [dρAvPC] =

(N−2K−2

)

K · (N

K

)N∑

i �=j

∫

�N

φK(αj − αi) d−α + O

(K2

N

)

= K − 1

N(N − 1)

N∑

i �=j

∫

�N

φK(αj − αi) dα + O

(K2

N

). (11)

ALAIN BOURGET

We now use the invariance of the function∑

i �=j φK(αj − αi) under the permu-tations in SN+1, the symmetric group of N + 1 elements, to replace the integrationover �N by the simpler integration over the unit cube [0, 1]N+1. As a consequence,we obtain

E�N [dρAvPC] = K − 1

N(N − 1)

N∑

i �=j

∫

[0,1]N+1φK(αj − αi) dα + O

(K2

N

)

= K − 1

N(N − 1)

N∑

i �=j

∫ 1

0

∫ 1

0φK(αj − αi) dαi dαj + O

(K2

N

)

= K

∫ 1

0

∫ 1

0φK(x − y) dy + O

(K2

N

)+ O

(1

K

).

We now make the change of variables u = x − y and v = x to get

E�N [dρAvPC] = K

[∫ 0

−1

∫ u+1

0φK(u) dv du +

∫ 1

0

∫ 1

u

φK(u) dv du

]

+ O

(K2

N

)+ O

(1

K

)

= K

[∫ 0

−1φK(u)(1 + u) du +

∫ 1

0φK(u)(1 − u) du

]

+ O

(K2

N

)+ O

(1

K

)

= K

∫ 1

−1φK(u)(1 − |u|) du + O

(K2

N

)+ O

(1

K

)

=∫ K

−K

φ(x) dx + O

(K2

N

)+ O

(1

K

)

=∫ b

a

φ(x) dx + O

(K2

N

)+ O

(1

K

).

This completes the proof of Theorem 1.1. �4. Proof of Theorem 1.2

In order to obtain the pointwise version of Theorem 1.1, we first show that thevariance of the pair correlation measure dρAv

PC is asymptotically small.

PROPOSITION 4.1. Under the same assumption of Theorem 1.1, we have that

Var�N [dρAvPC] = O

(K2

N

)+ O

(1

K

).


Proof. By definition of the variance, we need to prove

Var�N [dρAvPC] := E�N [(dρAv

PC)2] − (E�N [dρAvPC])2

= O

(K2

N

)+ O

(1

K

).

Consequently, it suffices to show that

E�N [(dρAvPC)2] = 1

σ 2(N, K)

∑

|l|=K

∑

|l′|=K

1

K2

K∑

i �=j

K∑

i′ �=j ′

×∫

�N

φK(θj (α; l) − θi(α; l))φK(θj ′(α; l′)

− θi′(α; l′)) d−α

=(∫ b

a

φ(x) dx

)2

+ O

(K2

N

)+ O

(1

K

). (12)

In order to prove (12), we use similar arguments as in the proof of Theorem 1.1.That is, we expand each of the functions φK(θj (α; l)−θi(α; l)) and φK(θj ′(α; l′)−θi′(α; l′)) in a first order Taylor series around αlj − αli and αl′

j ′ − αl′i′ . We get

E�N [(dρAvPC)2] = 1

σ 2(N, K)

∑

|l|=K

∑

|l′|=K

1

K2

K∑

i �=j

K∑

i′ �=j ′

×∫

�N

φK(αlj − αli )φK(αl′j ′ − αl′

i′ ) d−α

+ R2(φ, N, K).

The fact that the error term R2(φ, N, K) � K2/N follows directly from thebasic inequalities in (5) and the simple calculation

∫

�N

∣∣θj (α; l) − αj ||θi′(α; l′) − αl′i′ | d−α �

∫

�N

(αlj +1 − αlj )(αl′i′+1 − αl′

i′ ) d−α

= (l′i + 2)(lj + 3) − (l′i + 1)(lj + 3) − (l′i + 2)(lj + 2) + (l′i + 1)(lj + 2)

(N + 2)(N + 3)

= 1

(N + 2)(N + 3),

for all l′i , lj ∈ {1, . . . , N}.The rest of the proof follows closely the proof of Theorem 1.1. That is, we first

replace the sums over all configurations l and l′ by the sums over the configurationsl and l′ that contains at most one zero in each subintervals (αj , αj+1). This impliesthat

1

σ 2(N, K)

∑

|l|=K

∑

|l′|=K

1

K2

K∑

i �=j

K∑

i′ �=j ′

∫

�N

φK(αlj − αli )φK(αl′j ′ − αl′

i′ ) d−α

ALAIN BOURGET

= 1(N

K

)2

∑

|l|=K

∑

|l′|=K

1

K2

N∑

i �=j

N∑

i′ �=j ′

∫

�N

φK(αlj− αli

)φK(αl′j ′ − αl′

i′) d−α

+ O

(K2

N

)

=(N−2K−2

)2

(N

K

)2

N∑

i �=j

N∑

i′ �=j ′

∫

�N

φK(αj − αi)φK(αj ′ − αi′) dα + O

(K2

N

)

= (K − 1)2

(N(N − 1))2

N∑

i �=j

N∑

i′ �=j ′

∫

�N

φK(αj − αi)φK(αj ′ − αi′) dα + O

(K2

N

).

We now use the invariance of the function∑N

i �=j

∑Ni′ �=j ′ φK(αj − αi)φK(αj ′ −

αi′) under the elements of SN+1 to replace the integration over �N by [0, 1]N+1.Furthermore, we can assume without loss of generality that both i, j are differentfrom i ′, j ′ since the number of pair for which i ′ = i, or i ′ = j , or j ′ = i, or j ′ = j

is of the order N3. This yields

(K − 1)2

(N(N − 1))2

N∑

i �=j

N∑

i′ �=j ′

∫

�N

φK(αj − αi)φK(αj ′ − αi′) dα

= (K − 1)2

[∫ 1

0

∫ 1

0φK(αj − αi) dαj dαi

]

×[∫ 1

0

∫ 1

0φK(αj ′ − αi′) dαj ′ dαi′

]+ O

(K2

N

)

= (K − 1)2

K2

[∫ K

−K

φ(x) dx

][∫ K

−K

φ(x) dx

]+ O

(K2

N

)

=[∫ b

a

φ(x) dx

]2

+ O

(K2

N

)+ O

(1

K

).

This proves (12), and hence the proposition. �To complete the proof of Theorem 1.2, it suffices to apply Chebyshev’s in-

equality. Indeed, let h(x) be any increasing function such that h(x) → ∞ asx → ∞ and h2(K) = o(K) = O(

√N/K). If we denote by JN := {α ∈ �N :

|dρAvPC(φ;N, K, α) − ∫ b

aφ(x) dx| < 1/h(K)}, we then have

meas(�N/JN) � h2(K) · Var�N [dρAvPC]

= O

(h2(K)K2

N

)+ O

(h2(K)

K

).


The fact that these two terms go to zero as K → ∞ and N → ∞ follows fromour initial assumptions on h. By complementarity, we finally deduce that

meas(JN) = 1 + O

(h2(K)K2

N

)+ O

(h2(K)

K

). �

5. Pair Correlation Function

Clearly, both Theorems 1.1 and 1.2 remain valid when φ is merely a simple func-tion. Based on Rudnick and Sarnak’s paper [RS], one can also introduce the paircorrelation function for the zeros of Stieltjes polynomials as the counting function

R2([−s, s], α, K, N, l) := #

{1 � i �= j � K : |θ(l)

lj(α) − θ

(l)li

(α)| � s

K

}.

Consequently, for s > 0 and φ = χ[−s,s], the characteristic function of theinterval [−s, s], we obtain the following important corollary.

COROLLARY 5.1. Let K = o(N1/2) as N → ∞. For any s > 0, we have that

1

σ(N, K)K

∑

|l|=K

R2([−s, s], α, K, N, l) → 2s as K → ∞.

Acknowledgement

I would like to thank the referees for several insightful comments.

References

[B] BourgetA., A.: Nodal statistics for the Van Vleck polynomials, Comm. Math. Phys. 230(2002), 503–516.

[BT] Bourget, A. and Toth, J. A.: Asymptotic statistics of zeros for the Lamé ensemble, Comm.Math. Phys. 222 (2001), 475–493.

[G] Grünbaum, F.: Variations on a theme of Heine and Stieltjes: An electrostatic interpretationof the zeros of certain polynomials, J. Comput. Appl. Math. 99 (1998), 189–194.

[I] Ismail, M. E. H.: An electrostatic model for zeros of orthogonal polynomials, Pacific J.Math. 193 (2000), 355–369.

[MS] Martinez-Finkelshtein, A. and Saff, E. B.: Asymptotic properties of Van Vleck and Stieltjespolynomials, J. Approx. Theory 118(1) (2002), 131–151.

[RS] Rudnick, Z. and Sarnak, P.: The pair correlation function of fractional parts of polynomials,Comm. Math. Phys. 194 (1994), 61–70.

[St] Stieltjes, T. J.: Sur certains polynômes qui vérifient une équation différentielle linéaire dusecond ordre et sur la théorie des fonctions de Lamé, Acta Math. 8 (1885), 321–326.

[Sz] Szegö, G.: Orthogonal Polynomials, 3rd edn, Amer. Math. Soc., Providence, RI, 1967.[T] Toth, J. A.: The quantum C. Neuman problem, Internat. Math. Res. Notices 5 (1993), 137–

139.

ALAIN BOURGET

[V] Van Vleck, E. B.: On the polynomials of Stieltjes, Bull. Amer. Math. Soc. 4 (1898), 426–438.

[Vo] Volkmer, H.: Expansions in products of Heine–Stieltjes polynomials, Constr. Approx. 15(1999), 467–480.

[WW] Whittaker, E. T. and Watson, G. N.: A Course of Modern Analysis, 4th edn, CambridgeUniv. Press, Cambridge, 1963.


Generalized Bessel Functions andLie Algebra Representation

SUBUHI KHAN and GHAZALA YASMINDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, India.e-mail: [email protected]

(Received: 21 July 2004; in final form: 23 February 2005)

Abstract. The generalized Bessel functions (GBF) are framed within the context of the represen-tation Q(ω, m0) of the three-dimensional Lie algebra T3. The analysis has been carried out bygeneralizing the formalism relevant to Bessel functions. New generating relations and identitiesinvolving various forms of GBF are obtained. Certain known results are also mentioned as specialcases.

Mathematics Subject Classifications (2000): 33C10, 33C80, 33E20.

Key words: generalized Bessel functions, Lie group, Lie algebra, representation theory, generatingrelations.

1. Introduction

The theory of special functions plays an important role in the formalism of mathe-matical physics. Bessel functions (BF) [13], are among the most important specialfunctions with very diverse applications to physics, engineering and mathemat-ical analysis ranging from abstract number theory and theoretical astronomy toconcrete problems of physics and engineering.

The importance of BF has been further stressed by their various generalizations.Dattoli and his co-workers introduced and discussed generalized Bessel functions(GBF) and their multi-variable, multi-index extensions within purely mathematicaland applicative contexts (see, e.g., [2–8]). GBF have proved a powerful tool toinvestigate the dynamical aspects of physical problems such as electron scatteringby an intense linearly polarized laser wave, multi-photon processes and undulatorradiation. The analytical and numerical study of GBF has revealed their interestingproperties, which in some sense can be regarded as an extension of the propertiesof BF to a two-dimensional domain. In this connection, the relevance of GBF andtheir multi-variable extension in mathematical physics has been emphasized, sincethey provide analytical solutions to partial differential equations such as the multi-dimensional diffusion equation, the Schrödinger and Klein–Gordon equations. Thealgebraic structure underlying GBF can be recognized in full analogy with BF, thusproviding a unifying view to the theory of both BF and GBF.

300 SUBUHI KHAN AND GHAZALA YASMIN

A useful complement to the theory of GBF is offered by the introduction of 2-index 3-variable 1-parameter Bessel functions (2I3V1PBF), defined as ([8];p. 344(1,2))

Jm,n(x, y, z; ξ) =∞∑

s=−∞ξ sJm−s(x)Jn−s(y)Js(z), (1.1)

with the following generating function∞∑

m,n=−∞Jm,n(x, y, z; ξ)umvn

= exp

(x

2

(u − 1

u

)+ y

2

(v − 1

v

)+ z

2

(ξuv − 1

ξuv

)). (1.2)

The theory of special functions from the group-theoretic point of view is a well es-tablished topic, providing a unifying formalism to deal with the immenseaggregate of the special functions and a collection of formulas such as the relevantdifferential equations, integral representations, recurrence formulae, compositiontheorems, etc. ([15, 16]).

Within the group-theoretic context, indeed a given class of special functionsappears as a set of matrix elements of irreducible representations of a given Liegroup. The algebraic properties of the group are then reflected in the functionaland differential equations satisfied by a given family of special functions, whilstthe geometry of the homogeneous space determines the nature of the integral rep-resentation associated with the family.

The Bessel functions of integral order have been shown to be connected withthe faithful irreducible unitary representations of the real Euclidean group E3 in theplane ([14, 18]). The Euclidean group E3 is a real 3-parameter global Lie group,whose Lie algebra E3 has basis elements

J1 =( 0 0 1

0 0 00 0 0

), J2 =

( 0 0 00 0 10 0 0

), J3 =

( 0 −1 01 0 00 0 0

),

(1.3)

with commutation relations

[J1, J2] = 0, [J3, J1] = J2, [J3, J2] = −J1. (1.4)

For a theory of Bessel functions, it is sufficient to study the representation theoryof a three-dimensional complex local Lie group T3, which is the set of all 4 × 4matrices of the form

g =

1 0 0 τ

0 e−τ 0 c

0 0 eτ b

0 0 0 1

, b, c, τ ∈ C. (1.5)

GENERALIZED BESSEL FUNCTIONS AND LIE ALGEBRA REPRESENTATION 301

A basis for the Lie algebra T3 = L(T3) is provided by the matrices

J+ =

0 0 0 00 0 0 00 0 0 10 0 0 0

, J− =

0 0 0 00 0 0 10 0 0 00 0 0 0

,

(1.6)

J3 =

0 0 0 10 −1 0 00 0 1 00 0 0 0

,

with commutation relations

[J3, J+] = J+, [J3, J−] = −J−, [J+, J−] = 0. (1.7)

Further, we observe that the complex matrices

J+′ = −J2 + iJ1, J−′ = J2 + iJ1, J3′ = iJ3, i = √−1, (1.8)

satisfy the commutation relations identical with (1.7). Thus we say that T3 is thecomplexification of E3 and E3 is a real form of T3 ([10]). Due to this relationshipbetween T3 and E3, the abstract irreducible representation Q(ω, m0) of T3 ([12])induces an irreducible representation of E3.

In this paper, the authors derive generating relations involving GBF using therepresentation theory of the Lie group T3. In Section 2, we give a review of the basicproperties of 2I3V1PBF Jm,n(x, y, z; ξ) and their connections with other GBF andBF. In Section 3, we obtain the main results by relating 2I3V1PBF Jm,n(x, y, z; ξ)

to the representation Q(ω, m0) of the Lie algebra T3. In Section 4, we obtain var-ious new relations for the functions associated with 2I3V1PBF, also we mentionsome known relations. Finally, in Section 5, concluding remarks are given.

2. Properties of 2I3V1PBF Jm,n(x, y, z; ξ)

The 2I3V1PBF Jm,n(x, y, z; ξ) defined by Equations (1.1) and (1.2) satisfy thefollowing differential and pure recurrence relations:

∂

∂xJm,n(x, y, z; ξ) = 1

2 [Jm−1,n(x, y, z; ξ) − Jm+1,n(x, y, z; ξ)],∂

∂yJm,n(x, y, z; ξ) = 1

2 [Jm,n−1(x, y, z; ξ) − Jm,n+1(x, y, z; ξ)],(2.1)

∂

∂zJm,n(x, y, z; ξ) = 1

2

[ξJm−1,n−1(x, y, z; ξ) − 1

ξJm+1,n+1(x, y, z; ξ)

],

∂

∂ξJm,n(x, y, z; ξ) = z

2

[Jm−1,n−1(x, y, z; ξ) + 1

ξ 2Jm+1,n+1(x, y, z; ξ)

],


and

mJm,n(x, y, z; ξ) = x

2[Jm−1,n(x, y, z; ξ) + Jm+1,n(x, y, z; ξ)] +

+ z

2

[ξJm−1,n−1(x, y, z; ξ) + 1

ξJm+1,n+1(x, y, z; ξ)

],

(2.2)nJm,n(x, y, z; ξ) = y

2[Jm,n−1(x, y, z; ξ) + Jm,n+1(x, y, z; ξ)] +

+ z

2

[ξJm−1,n−1(x, y, z; ξ) + 1

ξJm+1,n+1(x, y, z; ξ)

].

The differential equations satisfied by Jm,n(x, y, z; ξ) are

− ∂2

∂x2− 1

x

∂

∂x− ξ

x2(2m − 1)

∂

∂ξ+ ξ 2

x2

∂2

∂ξ 2+ m2

x2− 1 = 0, (2.3)

− ∂2

∂y2− 1

y

∂

∂y− ξ

y2(2n − 1)

∂

∂ξ+ ξ 2

y2

∂

∂ξ 2+ n2

y2− 1 = 0. (2.4)

We note the following special cases of 2I3V1PBF Jm,n(x, y, z; ξ):

(1) Jm,n(x, y, z; 1) = Jm,n(x, y, z), (2.5)

where Jm,n(x, y, z) denotes 2-index 3-variable Bessel functions (2I3VBF) definedby the generating function ([5]; p. 3639(13)),

∞∑

m,n=−∞Jm,n(x, y, z)umvn

= exp

(x

2

(u − 1

u

)+ y

2

(v − 1

v

)+ z

2

(uv − 1

uv

)), (2.6)

(2) Jm,n(x, x, x; ξ) = Jm,n(x; ξ), (2.7)

where Jm,n(x; ξ) denotes 2-index 1-variable 1-parameter Bessel functions(2I1V1PBF) defined by the generating function ([5]; p. 3648(43)),

∞∑

m,n=−∞Jm,n(x; ξ)umvn

= exp

(x

2

((u − 1

u

)+

(v − 1

v

)+

(ξuv − 1

ξuv

))), (2.8)

(3) Jm,n(x, x, x; 1) = Jm,n(x), (2.9)

where Jm,n(x) denotes 2-index 1-variable Bessel functions (2I1VBF) defined bythe generating function ([5]; p. 3637(1)),

∞∑

m,n=−∞Jm,n(x)umvn

= exp

(x

2

((u − 1

u

)+

(v − 1

v

)+

(uv − 1

uv

))). (2.10)


(4) Jm,n(x, 0, 0; ξ) = Jm(x), (2.11)

where Jm(x) denotes Bessel functions (BF) defined by the generating function([13]; p. 113(4)),

∞∑

m=−∞Jm(x)um = exp

(x

2

(u − 1

u

)). (2.12)

The addition and multiplication theorems have noticeable relevance and areparticularly useful for numerical evaluation of 2I3V1PBF Jm,n(x, y, z; ξ). TheNeumann addition theorem for Jm,n(x, y, z; ξ) is given by

Jm,n(x ± p, y ± q, z ± r; ξ)

=∞∑

l,t=−∞Jm∓l,n∓t (x, y, z; ξ)Jl,t (p, q, r; ξ), (2.13)

and the multiplication theorem for Jm,n(x, y, z; ξ) is given in the following way:

Jm,n(λx, µy, µz; ξ)

= λmµn

∞∑

k,l=0

λkµl

k!l! Jm+k,n+l

(x, y, z; ξ

λ

)Hk,l

((1 − λ2

λ

)x

2,

(1 − µ2

µ

)y

2,

(1 − µ2

ξµ

)z

2

),

(2.14)

where Hk,l(x, y, z) are 2-index 3-variable Hermite polynomials defined by

Hk,l(x, y, z) =min {k,l}∑

p=0

(x/2)k−p(y/2)l−p(z/2)pk!l!p!(k − p)!(l − p)! . (2.15)

3. Representation Q(ω, m0) of T3 and Generating Relations

Miller ([12]) has determined realizations of the irreducible representaion Q(ω, m0)

of T3 where ω, m0 ∈ C such that ω �= 0 and 0 � Re m0 < 1. The spectrum S ofthis representation is the set {m0 + k : k an integer}, and the representation spaceV has a basis {fm : m ∈ S}, such that

J 3fm = mfm, J+fm = ωfm+1, J−fm = ωfm−1,

C0,0fm = (J+J−)fm = ω2fm, ω �= 0.(3.1)

The commutation relations satisfied by the operators J 3, J± are

[J 3, J+] = J+, [J 3, J−] = −J−, [J+, J−] = 0. (3.2)


In order to find the realizations of this representation on spaces of functions oftwo complex variables, x and y, Miller ([12]; pp. 59–60) has taken the functionsfm(x, y) = Zm(x)emy , such that Equations (3.1) are satisfied for all m ∈ S, wherethe differential operators J 3, J± are given by

J 3 = ∂

∂y,

J+ = ey

[∂

∂x− 1

x

∂

∂y

], (3.3)

J− = e−y

[− ∂

∂x− 1

x

∂

∂y

].

In particular, we are looking for the functions

fm,n(x, y, z, u, v; ξ) = Zm,n(x, y, z; ξ)umvn

such that

J 3fm,n = mfm,n, J+fm,n = ωfm+1,n, J−fm,n = ωfm−1,n,

C0,0fm,n = (J+J−)fm,n = ω2fm,n, ω �= 0,(3.4)

for all m ∈ S, and the operators J 3, J± satisfy the commutation relations (3.2).Again, we take the functions fm,n(x, y, z, u, v; ξ) = Zm,n(x, y, z; ξ)umvn such

that

J 3′fm,n = nfm,n, J+′

fm,n = ωfm,n+1, J−′fm,n = ωfm,n−1,

C0,0fm,n = (J+′J−′)fm,n = ω2fm,n, ω �= 0,(3.5)

for all n ∈ S, and the operators J 3′, J±′ satisfy the commutation relations identicalto (3.2).

First we assume that the set of linear differential operators J 3, J± take the form

J 3 = u∂

∂u,

J+ = u

[∂

∂x+ ξ

x

∂

∂ξ− u

x

∂

∂u

], (3.6)

J− = u−1

[− ∂

∂x+ ξ

x

∂

∂ξ− u

x

∂

∂u

],

and note that these operators satisfy the commutation relations (3.2).Further, we take the set of linear differential operators J 3′, J±′ as follows:

J 3′ = v∂

∂v,

J+′ = v

[∂

∂y+ ξ

y

∂

∂ξ− v

y

∂

∂v

], (3.7)

J−′ = v−1

[− ∂

∂y+ ξ

y

∂

∂ξ− v

y

∂

∂v

],


and note that these operators satisfy the commutation relations identical to (3.2).In terms of the functions Zm,n(x, y, z; ξ), relations (3.4) and (3.5) reduce to

(i)

[∂

∂x+ ξ

x

∂

∂ξ− m

x

]Zm,n(x, y, z; ξ) = ωZm+1,n(x, y, z; ξ),

(ii)

[− ∂

∂x+ ξ

x

∂

∂ξ− m

x

]Zm,n(x, y, z; ξ) = ωZm−1,n(x, y, z; ξ),

(iii)

[− ∂2

∂x2+ ξ 2

x2

∂2

∂ξ 2− 1

x

∂

∂x− ξ

x2(2m − 1)

∂

∂ξ+ m2

x2

]Zm,n(x, y, z; ξ)

= ω2Zm,n(x, y, z; ξ),

(3.8)

and

(i)

[∂

∂y+ ξ

y

∂

∂ξ− n

y

]Zm,n(x, y, z; ξ) = ωZm,n+1(x, y, z; ξ),

(ii)

[− ∂

∂y+ ξ

y

∂

∂ξ− n

y

]Zm,n(x, y, z; ξ) = ωZm,n−1(x, y, z; ξ),

(iii)

[− ∂2

∂y2+ ξ 2

y2

∂2

∂ξ 2− 1

y

∂

∂y− ξ

y2(2n − 1)

∂

∂ξ+ n2

y2

]Zm,n(x, y, z; ξ)

= ω2Zm,n(x, y, z; ξ),

(3.9)

respectively. The complex constant ω in these equations and in Equations (3.1) isclearly nonessential. Hence we will assume that ω = −1. For this choice of ω, andin terms of the functions Zm(x), relations (3.1) become ([12]; p. 60(3.25))

(i)

[d

dx− m

x

]Zm(x) = −Zm+1(x),

(ii)

[d

dx+ m

x

]Zm(x) = Zm−1(x),

(iii)

[− d2

dx2− 1

x

d

dx+ m2

x2

]Zm(x) = Zm(x).

(3.10)

We observe that (i) and (ii) of Equations (3.10) agree with the conventionalrecursion relations for BF Jm(x) and (iii) coincides with the differential equationfor Jm(x). Thus we see that Zm(x) = Jm(x) is a solution of Equations (3.10) forall m ∈ S.

Similarly, we see that for ω = −1, (iii) of Equations (3.8) and (3.9) coin-cide with the differential Equations (2.3) and (2.4), respectively, of 2I3V1PBFJm,n(x, y, z; ξ). In fact, for all m, n ∈ S the choice for Zm,n(x, y, z; ξ)

= Jm,n(x, y, z; ξ) satisfy Equations (3.8) and (3.9). It follows from the above dis-cussion that the functions fm,n(x, y, z, u, v; ξ) = Jm,n(x, y, z; ξ)umvn, m, n ∈ S

form a basis for a realization of the representation Q(−1, m0) of T3. By using ([12];


p. 18 (Theorem 1.10)), this representation of T3 can be extended to a local multi-plier representation ([12], p. 17) of T3. Using operators (3.6), the local multiplierrepresentation T (g), g ∈ T3 defined on F , the space of all functions analytic ina neighbourhood of the point (x0, y0, z0, u0, v0; ξ 0) = (1, 0, 0, 1, 1, 1), takes theform

[T (exp τJ3)f ](x, y, z, u, v; ξ) = f (x, y, z, eτ u, v; ξ),

[T (exp cJ−)f ](x, y, z, u, v; ξ)

= f

(x

(1 − 2c

ux

)1/2

, y, z, u

(1 − 2c

ux

)1/2

, v; ξ

(1 − 2c

ux

)−1/2), (3.11)

[T (exp bJ+)f ](x, y, z, u, v; ξ)

= f

(x

(1 + 2bu

x

)1/2

, y, z, u

(1 + 2bu

x

)−1/2

, v; ξ

(1 + 2bu

x

)1/2).

If g ∈ T3 has parameters (b, c, τ ), then

T (g) = T (exp bJ+)T (exp cJ−)T (exp τJ3),

and therefore we obtain

[T (g)f ](x, y, z, u, v; ξ)

= f

(x

(1 − 2c

ux

)1/2(1 + 2bu

x

)1/2

,

y, z, eτ u

(1 − 2c

ux

)1/2(1 + 2bu

x

)−1/2

,

v; ξ

(1 − 2c

ux

)−1/2(1 + 2bu

x

)1/2),

∣∣∣∣2bu

x

∣∣∣∣ < 1,

∣∣∣∣2c

ux

∣∣∣∣ < 1.

(3.12)

The matrix elements of T (g) with respect to the analytic basis (fm,n)m,n∈S arethe functions Alk(g) uniquely determined by Q(−1, m0) of T3, and we obtain therelations

[T (g)fm0+k,n](x, y, z, u, v; ξ) =∞∑

l=−∞Alk(g)fm0+l,n(x, y, z, u, v; ξ),

(3.13)k = 0, ±1, ±2, ±3, . . . ,

which simplifies to the identity

emτ

(1 − (2c/ux)

1 + (2bu/x)

)m/2

Jm,n

(x

(1 − 2c

ux

)1/2(1 + 2bu

x

)1/2

,

y, z; ξ

(1 − 2c

ux

)−1/2(1 + 2bu

x

)1/2)


=∞∑

l=−∞Al,m−m0(g)Jm0+l,n(x, y, z; ξ)um0+l−m, (3.14)

and the matrix elements Alk(g) are given by ([12]; p. 56(3.12)′),

Al,m−m0(g) = emτ (−1)|p|

|p|! c(−p+|p|)/2b(p+|p|)/20F1(−; |p| + 1; bc), (3.15)

where 0F1 denotes confluent hypergeometric functions ([13]). Substituting (3.15)into (3.14), we obtain the first main generating relation

(1 − (2c/ux)

1 + (2bu/x)

)m/2

Jm,n

(x

(1 − 2c

ux

)1/2(1 + 2bu

x

)1/2

,

y, z; ξ

(1 − 2c

ux

)−1/2(1 + 2bu

x

)1/2)

=∞∑

p=−∞

(−1)|p|

|p|! c(−p+|p|)/2b(p+|p|)/20F1(−; |p| + 1; bc)×

× Jm+p,n(x, y, z; ξ)up,

∣∣∣∣2bu

x

∣∣∣∣ < 1,

∣∣∣∣2c

ux

∣∣∣∣ < 1. (3.16)

Similarly for the operators (3.7), we obtain the second main generating relation(

1 − (2c′/vy)

1 + (2b′v/y)

)n/2

Jm,n

(x, y

(1 − 2c′

vy

)1/2(1 + 2b′v

y

)1/2

,

z; ξ

(1 − 2c′

vy

)−1/2(1 + 2b′v

y

)1/2)

=∞∑

q=−∞

(−1)|q|

|q|! c′(−q+|q|)/2b′(q+|q|)/20F1(−; |q| + 1; bc)×

× Jm,n+q(x, y, z; ξ)vq,

∣∣∣∣2b′vy

∣∣∣∣ < 1,

∣∣∣∣2c′

vy

∣∣∣∣ < 1. (3.17)

Further, if bc �= 0, we can introduce the coordinates r, ν such that b = rν/2and c = −(r/2ν), with these new coordinates generating relation (3.16) becomes

(1 + (r/uνx)

1 + (rνu/x)

)m/2

Jm,n

(x

(1 + r

uνx

)1/2(1 + rνu

x

)1/2

,

y, z; ξ

(1 + r

uνx

)−1/2(1 + rνu

x

)1/2)

=∞∑

p=−∞(−ν)pJp(r)Jm+p,n(x, y, z; ξ)up,

∣∣∣∣r

uνx

∣∣∣∣ < 1,

∣∣∣∣rνu

x

∣∣∣∣ < 1,

(3.18)


which, for u = 1, reduces to a generalization of Graf’s addition theorem ([9];p. 44).

Also, if b′c′ �= 0, we can introduce the coordinates r ′, ν ′ such that b′ = r ′ν ′/2and c′ = −(r ′/2ν ′). In this case, generating relation (3.17) becomes

(1 + (r ′/vν ′y)

1 + (r ′ν ′v/y)

)n/2

Jm,n

(x, y

(1 + r ′

vν ′y

)1/2(1 + r ′ν ′v

y

)1/2

,

z; ξ

(1 + r ′

vν ′y

)−1/2(1 + r ′ν ′v

y

)1/2)

=∞∑

q=−∞(−ν ′)qJq(r

′)Jm,n+q(x, y, z; ξ)vq,

∣∣∣∣r ′

vν ′y

∣∣∣∣ < 1,

∣∣∣∣r ′ν ′v

y

∣∣∣∣ < 1.

(3.19)

4. Applications

We discuss some applications of the generating relations obtained in the precedingsection.

(I) Taking c = 0 and u = 1 in generating relation (3.16), we obtain

(1 + 2b

x

)−m/2

Jm,n

(x

(1 + 2b

x

)1/2

, y, z; ξ

(1 + 2b

x

)1/2)

=∞∑

p=0

(−b)p

p! Jm+p,n(x, y, z; ξ),

∣∣∣∣2b

x

∣∣∣∣ < 1. (4.1)

Again taking b = 0 and u = 1 in generating relation (3.16), we get

(1 − 2c

x

)m/2

Jm,n

(x

(1 − 2c

x

)1/2

, y, z; ξ

(1 − 2c

x

)−1/2)

=∞∑

p=0

(−c)p

p! Jm−p,n(x, y, z; ξ),

∣∣∣∣2c

x

∣∣∣∣ < 1. (4.2)

Further, taking y = z = 0 in generating relations (4.1) and (4.2), we obtain theformulas of Lommel ([12], p. 62(3.30,3.31)) respectively. Similarly, we can obtainresults corresponding to generating relation (3.17).

(II) Taking ξ = u = ν = 1 in generating relation (3.18), we obtain the followingresult:

Jm,n

(x

(1 + r

x

), y, z

)=

∞∑

p=−∞Jp(r)Jm+p,n(x, y, z),

∣∣∣∣r

x

∣∣∣∣ < 1, (4.3)


where Jm,n(x, y, z) is given by Equation (2.6). Similar results can be obtained fromgenerating relations (3.16), (3.17) and (3.19).

(III) Taking y = z = x in generating relation (3.18), we get(

1 + (r/uνx)

1 + (ruν/x)

)m/2

Jm,n

(x

(1 + r

uνx

)1/2(1 + ruν

x

)1/2

,

x, x;(

1 + r

uνx

)−1/2(1 + ruν

x

)1/2)

=∞∑

p=−∞(−ν)pJp(r)Jm+p,n(x; ξ)up,

∣∣∣∣r

uνx

∣∣∣∣ < 1,

∣∣∣∣ruν

x

∣∣∣∣ < 1, (4.4)

where Jm,n(x; ξ) is given by Equation (2.8).Further, taking ξ = u = ν = 1 in generating relation (4.4), we obtain

Jm,n

(x

(1 + r

x

))=

∞∑

p=−∞Jp(r)Jm+p,n(x),

∣∣∣∣r

x

∣∣∣∣ < 1, (4.5)

where Jm,n(x) is given by Equation (2.10).Similarly, we can obtain results corresponding to generating relations (3.16),

(3.17) and (3.19).

(IV) Taking y = z = 0 in generating relation (3.16), we obtain ([12], p. 62(3.29))(

1 − (2c/ux)

1 + (2bu/x)

)m/2

Jm

(x

(1 − 2c

ux

)1/2(1 + 2bu

x

)1/2)

=∞∑

p=−∞

(−1)|p|

|p|! c(−p+|p|)/2b(p+|p|)/20F1(−; |p| + 1; bc)Jm+p(x)up,

∣∣∣∣2bu

x

∣∣∣∣ < 1,

∣∣∣∣2c

ux

∣∣∣∣ < 1, (4.6)

where Jm(x) is given by Equation (2.12).Again, taking y = z = 0 and u = 1 in generating relation (3.18), we get a

generalization of Graf’s addition theorem ([12], p. 63(3.32))(

1 + (r/νx)

1 + (rν/x)

)m/2

Jm

(x

(1 + r

νx

)1/2(1 + rν

x

)1/2)

=∞∑

p=−∞(−ν)pJp(r)Jm+p(x),

∣∣∣∣rν

x

∣∣∣∣ < 1,

∣∣∣∣r

νx

∣∣∣∣ < 1. (4.7)

Similar results can be obtained from generating relations (3.17) and (3.19)respectively.


5. Concluding Remarks

We note that the expressions (3.13) are valid only for group elements g in a suffi-ciently small neighbourhood of the identity element of the Lie group T3. However,we can also use the operators (3.6) to derive generating relations for 2I3V1PBFand related functions with group elements bounded away from the identity.

If f (x, y, z, u, v; ξ) is a solution of the equation C0,0f = ω2f , i.e.,(

− ∂2

∂x2− 1

x

∂

∂x− ξ

x2(2m − 1)

∂

∂ξ+ ξ 2

x2

∂2

∂ξ 2+ m2

x2

)f (x, y, z, u, v; ξ),

= ω2f (x, y, z, u, v; ξ), (5.1)

then the function T (g)f given by (3.12) satisfies the equation

C0,0(T (g)f ) = ω2(T (g)f ).

This follows from the fact that C0,0 commutes with the operators J+, J− andJ 3. Now if f is a solution of the equation

(x1J+ + x2J

− + x3J3)f (x, y, z, u, v; ξ) = λf (x, y, z, u, v; ξ), (5.2)

for constants x1, x2, x3 and λ, then T (g)f is a solution of the equation

[T (g)(x1J+ + x2J

− + x3J3)T (g−1)][T (g)f ] = λ[T (g)f ]. (5.3)

The inner automorphism µg of Lie group T3 defined by

µg(h) = ghg−1, h ∈ T3, (5.4)

induces an automorphism µ�g of Lie algebra T3, where

µ�g(α) = gαg−1, α ∈ T3.

If α = x1J+ + x2J

− + x3J3, where J+, J− and J3 are given by Equation (1.6)

and g is given by Equation (1.5), then we have

µ�g(α) = (x1eτ − bx3)J

+ + (x2e−τ + cx3)J− + x3J

3, (5.5)

as a consequence of which, we can write

T (g)(x1J+ + x2J

− + x3J3)T (g−1)

= (x1eτ − bx3)J+ + (x2e−τ + cx3)J

− + x3J3. (5.6)

To give an example of the application of these remarks, we consider the functionf (x, y, z, u, v; ξ) = Jm,n(x, y, z; ξ)umvn, m ∈ C . Since C0,0f = f and J 3f =mf , so the function

[T (g)f ](x, y, z, u, v; ξ)

= emτ

(u2 − (2cu/x)

1 + (2bu/x)

)m/2

vnJm,n

((x + 2bu)1/2

(x − 2c

u

)1/2

, y, z; ξ ×

×(

1 − 2c

ux

)−1/2(1 + 2bu

x

)1/2), (5.7)


satisfies the equations

C0,0[T (g)f ] = [T (g)f ], (5.8)

(−bJ+ + cJ− + J 3)[T (g)f ] = m[T (g)f ]. (5.9)

For τ = b = 0 and c = −1, we can express the function (5.7) in the form

h(x, y, z, u, v; ξ)

=(

u2 + 2u

x

)m/2

vnJm,n

((x2 + 2x

u

)1/2

, y, z; ξ

(1 + 2

ux

)−1/2). (5.10)

Now using the Laurent expansion

h(x, y, z, u, v; ξ) =∞∑

k=−∞hk,n(x, y, z; ξ)ukvn, |xu| < 2,

in Equation (5.8), we note that hk,n(x, y, z; ξ) is a solution of differential equation(2.3) for each integer k. Since the function h(x, y, z, u, v; ξ) is bounded for x =y = z = 0, we have

hk,n(x, y, z; ξ) = ckJk,n(x, y, z; ξ), ck ∈ C.

Thus

hk,n(x, y, z, u, v; ξ) =∞∑

k=−∞ckJk,n(x, y, z; ξ)ukvn. (5.11)

Now from Equation (5.9), we have

(−J− + J 3)h(x, y, z, u, v; ξ) = mh(x, y, z, u, v; ξ)

and therefore it follows that

ck+1 = (m − k)ck.

Further, taking x = y = z = 0 in (5.10), and using (5.11), we get c0 =1/(m + 1) and, hence, ck = 1/(m − k + 1). Thus we obtain the followingresult:

(u2 + 2u

x

)m/2

Jm,n

((x2 + 2x

u

)1/2

, y, z; ξ

(1 + 2

ux

)−1/2)

=∞∑

k=−∞

Jk,n(x, y, z; ξ)uk

(m − k + 1), |xu| < 2, (5.12)

which is obviously not a special case of generating relation (3.16).The result (5.12) was obtained by using operators (3.6). We can obtain another

result by using operators (3.7). Several other examples of generating relations canbe derived by this method, see, e.g., Weisner [17].


The theory of BF is rich and wide, and certainly provides an inexhaustible fieldof research. A large number of functions are recognized as belonging to the BFfamily. Many variable BF were introduced at the beginning of the last century,see, e.g., [1, 11], forgotten for many years and reconsidered within the context ofvarious physical applications at the end of the last century, see, e.g., [2–8].

We have considered GBF within the group representation formalism. The2I3V1PBF Jm,n(x, y, z; ξ) appeared as basis functions for a realization of the rep-resentation Q(−1, m0) of the Lie algebra T3. The analysis presented in thispaper confirms the possibility of extending this approach to other useful formsof GBF.

Acknowledgement

The authors are thankful to the anonymous referee for valuable suggestions forimproving the presentation of the paper.

References

1. Appell, P.: Sur l’inversion approchée de certains integrales realles et sur l’expansion del’equation de Kepler et des fonctions de Bessel, CR Acad. Sci. Paris Sér. I Math. 160 (1915),419–423.

2. Chiccoli, C., Dattoli, G., Lorenzutta, S., Maino, G. and Torre, A.: Theory of one parametergeneralized Bessel functions, Monograph, Gruppo Nazionale Informatica Mathematica CNR,Rome, 1992.

3. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G., Richetta, M. and Torre, A.: Generatingfunctions of multi-variable generalized Bessel functions and Jacobi-elliptic functions, J. Math.Phys. 33 (1992), 25–36.

4. Dattoli, G., Giannessi, L., Mezi, L. and Torre, A.: Theory of generalized Bessel functions,Nuovo Cimento Soc. Ital. Fis. B (12) 105 (1990), 327–343.

5. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A., Voykov, G. and Chiccoli, C.: Theory of twoindex Bessel functions and applications to physical problems, J. Math. Phys. 35(7) (1994),3636–3649.

6. Dattoli, G., Renieri, A. and Torre, A.: Lectures on the free electron laser theory and relatedtopics, Singapore, 1993.

7. Dattoli, G., Torre, A., Lorenzutta, S., Maino, G. and Chiccoli, C.: Theory of generalized Besselfunctions II, Nuovo Cimento Soc. Ital. Fis. B (12) 106 (1991), 21–32.

8. Dattoli, G. and Torre, A.: Theory and Applications of Generalized Bessel Functions, ARACNE,Rome, Italy, 1996.

9. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: Higher TranscendentalFunctions, Vol. II, McGraw Hill, New York, 1953.

10. Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.11. Jekhowsky, B.: Les fonctions de Bessel de plusieurs variables exprimées pour les fonctions de

Bessel d’une variable, CR Acad. Sci. Paris Sér. I Math. 162 (1916), 38–319.12. Miller, W., Jr.: Lie Theory and Special Functions, Academic Press, New York, 1968.13. Rainville, E. D.: Special Functions, Macmillan, New York, 1960.14. Vilenkin, N. Y.: Bessel functions and representations of the group of Euclidean motion, Uspekhi

Mat. Nauk [N.S.] 11(3) (1956), 69–112.


15. Vilenkin, N. Y.: Special Functions and the Theory of Group Representation, Amer. Math. Soc.,Providence, RI, 1968.

16. Wawrzynczyk, A.: Group Representation and Special Functions, PWN-Polish Scientific Publ.,Warsaw, 1984.

17. Weisner, L.: Generating functions for Bessel functions, Canad. J. Math. 11 (1959), 148–155.18. Wigner, E. P.: The application of group theory to the special functions of mathematical physics,

Princeton Lecture Notes, 1955.

Mathematical Physics, Analysis and Geometry (2005) 8: 315–360 © Springer 2006DOI: 10.1007/s11040-005-2970-x

The Band-Edge Behavior of the Densityof Surfacic States

WERNER KIRSCH1 and FRÉDÉRIC KLOPP2

1Fakultät für Mathematik and SFB-TR 12, Ruhr Universität Bochum, D-44780 Bochum, Germany.e-mail: [email protected], UMR 7539 CNRS, Institut Galilée, Université de Paris-Nord, 99 venue J.-B. Clément,F-93430 Villetaneuse, France. e-mail: [email protected]

(Received: 27 July 2004; in final form: 25 February 2005)

Abstract. This paper is devoted to the asymptotics of the density of surfacic states near the spectraledges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered:fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuat-ing edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably‘reduced’ to the surface. For stable edges, the surface density of states behaves like the surface densityof states of a constant (equal to the expectation of the random potential) surface potential. Amongthe tools used to establish this are the asymptotics of the surface density of states for constant surfacepotentials.

Mathematics Subject Classifications (2000): 35P20, 46N50, 47B80.

Key words: random Schrödinger operators, density of states, Lifshits tails, surface states.

0. Introduction

On Zd (d = d1 + d2, d1 > 0, d2 > 0), we consider random Hamiltonians of the

form Hω = − 12� + Vω, where

• −� is the free Laplace operator, i.e., −(�u)(n) = ∑|m−n|=1 u(m);

• Vω is a random potential concentrated on the sublattice Zd1 × {0} ⊂ Z

d of theform

Vω(γ1, γ2) ={

ωγ1 if γ2 = 0,0 if γ2 �= 0,

γ = (γ1, γ2) ∈ Zd1 × Z

d2 = Zd (0.1)

and (ωγ1)γ1∈Zd1 is a family of i.i.d. bounded random variables. For the sake of

simplicity, let us assume that the random variables are uniformly distributedin [a, b] (a < b).

Above as well as for the rest of this paper we use the max-norm |n| = max{|xi |;1 � i � d} on the lattice Z

d (resp. Zd1 , etc.).

316 WERNER KIRSCH AND FREDERIC KLOPP

To keep the exposition as simple as possible in the introduction, we use thesequite restrictive assumptions. We will deal with more general models in the nextsection.

The operator Hω is bounded for almost every ω. It is ergodic with respect toshifts parallel to the surface. So we know there exists � the almost sure spectrumof Hω (see, e.g., [14, 23]).

For Hω, one defines the density of surface states (the DSS in the sequel), say dns ,in the following way (see, e.g., [2, 3, 8, 20]): for ϕ ∈ C∞

0 (R), we set

(ϕ, dns) = E(tr(�1[ϕ(Hω) − ϕ(− 12�)]�1)), (0.2)

where �1 is the orthogonal projector on the subspace Cδ0 ⊗ �2(Zd2) ⊂ �2(Zd).Here, δ0 denotes the vector with components (δ0j )j∈Z

d1 .Obviously, Equation (0.2) defines the integrated density of surface states ns only

up to a constant. We choose this constant so that ns vanishes below �. Note that,if �0 denotes the spectrum of − 1

2�, one has �0 ⊂ �. We will see later on that, upto addition of a well controlled distribution, ns is a positive measure.

One knows that � = σ(− 12�) ∪ supp(dns) (see [2, 8, 9]). We will study the

behavior of ns at the edges of �. To simplify this set as much as possible, we willassume that the support of the random variables (ωγ1)γ1∈Z

d1 is connected, say it isthe interval [a, b]. Under this assumption, we know that

LEMMA 0.1. � is a compact interval given by

� = σ(− 12�d1) +

⋃

t∈[a,b]σ(− 1

2�d2 + t�20), (0.3)

where �20 is the projector on the unit vector δ2

0 ∈ �2(Zd2).

This is a consequence of a standard characterization of � in terms of periodicpotentials (see [14, 23]). The assumption that the random variables have connectedsupport can be relaxed; more connected components for the support of the randomvariables will in general give rise to more spectral edges (as in the case of bulkrandomness, see [16]). For the value of �, two different possibilities occur:

(1) � = σ(− 12�) + [−α, β] = [−d − α, d + β] where α = α(a), β = β(b) and

α + β > 0; this occurs

• if d2 � 2 and either a < 0, in which case α(a) > 0, or b > 0, in which caseβ(b) > 0,

• if d2 � 3 and a > a0 or b > b0, where, by (0.3), the thresholds a0 and b0

are uniquely determined by the family of operators (− 12�d2 + t�2

0)t∈R.

If α > 0 (resp. β > 0), we say that the left (resp. right) edge is a ‘fluctuationedge’ or ‘fluctuation boundary’ (see [23]). If α = 0 (resp. β = 0), we willspeak of a ‘stable edge’ or ‘stable boundary’.

ASYMPTOTICS OF THE DENSITY OF SURFACIC STATES 317

(2) � = σ(− 12�); this occurs only in d2 � 3 and if a is not too large, that is, if

a ∈ (0, a0].In this case, both spectral edges are stable.

On the other hand, it is well known (see [24]) that,

• if d2 = 1, 2, then, for a > 0, σ(− 12�d2 − a�2

0) = [−d2, d2] ∪ {λ(a)}, andthe spectrum in [−d2, d2] is purely absolutely continuous and λ(a) is a simpleeigenvalue;

• if d2 � 3, there exists a0 > 0 such that

– if 0 < a < a0, then, σ(− 12�d2 − a�2

0) = [−d2, d2], and the spectrum ispurely absolutely continuous;

– if a = a0, then

∗ if d2 = 3, 4, then σ(− 12�d2 − a�2

0) = [−d2, d2], the spectrum is purelyabsolutely continuous, and −d2 is a resonance for − 1

2�d2 − a�20;

∗ if d2 � 5, then σ(− 12�d2 − a�2

0) = [−d2, d2], the spectrum is purelyabsolutely continuous in [−d2, d2), and −d2 is a simple eigenvalue for− 1

2�d2 − a�20;

– if a > a0, then, σ(− 12�d2 −a�2

0) = [−d2, d2]∪{λ(a)}, and the spectrum in[−d2, d2] is purely absolutely continuous and λ(a) is a simple eigenvalue;

For the operator − 12�d2 + b�2

0, we have a symmetric situation.Our aim is to study the density of surface states near the edges of �. In the

present case, both edges are obviously symmetric. So we will only describe thelower edge. One has to distinguish between the case of fluctuation and stable edges.The behavior in the two cases are radically different.

0.1. THE STABLE EDGE

As the discussion for lower and upper edge are symmetric, let us assume the loweredge is stable and work near that edge.

In the case of a stable edge, it is convenient to modify the normalization of theDSS. Therefore, we introduce the operator

Ht = − 12� + t1 ⊗ �2

0.

As above, let a be the infimum of the random variables (ωj )j . For ϕ ∈ C∞0 (R),

define

(ϕ, dns,norm) = E(tr(�1[ϕ(Hω) − ϕ(Ha)]�1)).

The advantage of this renormalization is that the DSS ns,norm is the distributionalderivative of a positive measure. Indeed, for ϕ ∈ C∞

0 (R), define

(ϕ, dNs,norm) = −E(tr(�1[P(ϕ)(Hω) − P(ϕ)(Ha)]�1)),


where

P(ϕ)(x) =∫ +∞

x

ϕ(t) dt.

Clearly, dNs,norm is independent of the anti-derivative of ϕ chosen to define it; it isa positive measure and

ns,norm = − d

dENs,norm = −dNs,norm.

Let nts be the IDSS for Ht . As above, one can define a anti-derivative of nt

s ; denoteit by −Nt

s . Let nts,norm be the normalized version of nt

s , i.e. nts,norm = nt

s − nas . One

has

ns,norm + nas = ns. (0.4)

One problem one encounters when studying ns is that very little is known about itsregularity for random surfacic models (see, nevertheless, [21]). Thanks to (0.4), weknow that ns is the difference of two distributions each of which is the derivative ofa signed measure. So we can take the counting function of dNs as dNs = dNs,norm+dNa

s is the sum of two measures. Thus, we define its counting function

Ns(E) =∫ E

−∞dNs(e). (0.5)

An obvious consequence of (0.4) is the

PROPOSITION 0.1. One has

Nas (E) � Ns(E) � Nb

s (E). (0.6)

This inequality is useful only at certain types of (stable) spectral edges, seeSection 0 for details.

In Section 5.1, we study the asymptotics for Nts . As a consequence of this study,

we prove

THEOREM 0.1. Assume d2 = 1 or 2. Then one has

Ns(E) ∼E→−dE>−d

Vol(Sd1−1)

d1(d1 + 2)(2π)d1· (E + d)1+d1/2 if d2 = 1,

2Vol(Sd1−1)

d1(d1 + 2)(2π)d1· (E + d)1+d1/2

| log(E + d)| if d2 = 2,

where Sd1−1 is the d1 − 1-dimensional unit sphere.


If a > 0, this result is an immediate consequence of Proposition 0.1 and ofTheorem 1.1 giving the asymptotics of the IDSS for constant surface potential (seealso Section 5.1). If a = 0, one needs to improve upon (0.6) as the left-hand sideof this inequality vanishes making it unusable. This is the purpose of Theorem 1.2.

When d2 � 3, the situation becomes more complicated and we are only able touse Proposition 0.1 to get the two-sided estimate

Ca(1 + o(1))

(1 + aI)� (2π)d

s(E + d)(E + d)1+d1/2· Ns(E) � C

b(1 + o(1))

(1 + bI), (0.7)

where C is a positive constant depending only on the dimensions d1 and d2 (seeSection 5.1) and

s(x) = 1

2|x|(d2−2)/2,

I = 1

2sup

θ1∈Td1

∫

θ2∈Td2

(

d −d1∑

j=1

cos(θj

1 ) −d2∑

j=1

cos(θj

2 )

)−1

dθ2.

Here, and in the sequel, the measure dθα (α ∈ {1, 2}) is the Haar measure on thetorus T

dα , i.e. the Lebesgue measure normalized to have total mass equal to one.Let us note that, if a < 0 < b, the inequality (0.7) does not give much

information of the actual behavior of Ns(E) when d2 � 3.

0.2. THE FLUCTUATION EDGE

Here, we assume that E0 = inf σ(Hω) is strictly below −d = inf σ(− 12�). In this

case, E0 is a fluctuation edge of the spectrum.Below the spectrum of − 1

2�, the density of surface states ns is positive; hence,it is a Borel measure and the integrated density of surface states Ns(E) can bedefined as its distribution function, i.e. Ns(E) = ns((−∞, E)) for E < −d. Wewill prove Lifshitz type behavior for Ns(E) for E ↘ E0 which is characteristicfor fluctuation edges. However, the Lifshitz exponent, in the homogeneous casetypically equal to −(d/2), is given by −(d1/2) in our case. More precisely, we willshow

limE↘E0

ln | ln(Ns(E))|ln(E − E0)

= −d1

2.

1. The Main Results

Let us now describe the general model we consider. Let H be a translational invari-ant Jacobi matrix with exponential off-diagonal decay that is H = ((hγ−γ ′))γ,γ ′∈Zd

such that,


(H0.a): h−γ = hγ for γ ∈ Zd and for some γ �= 0, hγ �= 0.

(H0.b): There exists c > 0 such that, for γ ∈ Zd ,

|hγ | � 1

ce−c|γ |.

The infinite matrix H defines a bounded self-adjoint operator on �2(Zd). Using theFourier transform, it is easily seen that H is unitarily equivalent to the multiplica-tion by the function θ �→ h(θ) defined by

h(θ) =∑

γ∈Z

hγ eiγ θ , where θ = (θ1, . . . , θd),

acting as an operator on L2(Td) where Td = R

d/(2πZd) (the Lebesgue measure

on Td is normalized so that the constant function 1 has norm 1). The function

h is real analytic on Td . We normalize it so that it be nonnegative and 0 be its

minimum.As both ends of the spectrum of our operator play symmetric parts, we only

study what happens at a left edge, i.e. near the bottom of the spectrum. All ourassumptions will reflect this fact.

1.1. THE CASE OF A CONSTANT SURFACE POTENTIAL

We will start with a study of the density of surface states when the surfacic potentialV is constant, i.e. V = t�2

0. We define the operator Ht = H + t1 ⊗ �20. We

prove two results on Ht . The first one is a criterion for the positivity of Ht and adescription of its infimum when it is negative; the other result describes the densityof surface states near 0 when Ht is nonnegative.

In the present section, we assume

(H1): the function h: Td → R admits a unique minimum; i.e. its Hessian is

nondegenerate.

If H is − 12�, then h = h0 where

h0(θ) := cos(θ1) + · · · + cos(θd). (1.1)

In this case, assumption (H1) is satisfied. Below, we give an example why consid-ering more general Hamiltonians can be of interest.

For the sake of definiteness, we assume the minimum of h to be 0. This amountsto adding a constant to H .

We start with a characterization of the infimum of the spectrum of Ht . There-fore, write h(θ) = h(θ1, θ2) where θ = (θ1, θ2), θ1 ∈ T

d1 , θ2 ∈ Td2 . Define

I (θ1, z) =∫

Td2

1

h(θ1, θ2) − zdθ2. (1.2)


We recall that the measures dθ2 is normalized so that the measure of Td2 be equal

to 1.We prove

PROPOSITION 1.1. Assume (H0) is satisfied.Ht is nonnegative if and only if t satisfies

1 + tI∞ � 0 where I∞ := supθ1∈T

d1

∫

Td2

1

h(θ1, θ2)dθ2. (1.3)

Assume now that 1 + tI∞ < 0. Then, there exists a unique E0 ∈ (−∞, 0] such that

∀θ1 ∈ Td1, 1 + tI (θ1, E0) � 0 and ∃θ1 ∈ T

d1, 1 + tI (θ1, E0) = 0.

Moreover, E0 is the infimum of the spectrum of Ht .

Proposition 1.1 is proved in Section 5.Criterion (1.3) immediately gives the obvious fact that if t � 0 then Ht is

nonnegative. As we assumed that h has only nondegenerate minima, if d2 = 1, 2and t < 0, then Ht is not nonnegative.

We now turn to our second result. It describes the asymptotics of Nts near 0

when (1.3) is satisfied. Recall that Nts is the density of surface states of Ht .

THEOREM 1.1. Assume t satisfies condition (1.3). Define

I =∫

Td2

1

h(0, θ2)dθ2.

One has

• if d2 = 1:∫ E

0dNt

s (e) ∼E→0+

Vol(Sd1−1)

d1(d1 + 2)(2π)d1

√Det(Q1 − RQ−1

2 R∗)· E1+d1/2,

• if d2 = 2:∫ E

0dNt

s (e) ∼E→0+

2Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)

E1+d1/2

| log E| .

If d2 � 3 and 1 + t · I > 0, then, one has

∫ E

0dNt

s (e) ∼E→0+

c(d1, d2)Vol(Sd2−1)Vol(Sd1−1)

d(2π)d√

DetQ· t

1 + tI· s(E)E1+d1/2.

If d2 � 3 and 1 + t · I = 0, if in addition we assume that θ1 �→ I (θ1, 0) :=∫T

d2 (h(θ1, θ2))−1 dθ2 has a local maximum for θ1 = 0, then one has


• if d2 = 3:∫ E

0dNt

s (e) de ∼E→0+

∫|θ1|�1 Arg(−i|1 − θ2

1 |1/2 + g(θ1)) dθ1

d1(d1 + 2)π(2π)d1


2 R∗)· E1+d1/2,

• if d2 = 4:∫ E

0dNt

s (e) ∼E→0+ − 2Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)

E1+d1/2

| log E| ,

• if d2 � 5:∫ E

0dNt

s (e) ∼E→0+

c(d1, d2)Vol(Sd2−1)Vol(Sd1−1)

d(2π)d√

DetQ· −1

J· s(E)Ed1/2.

Here, we used the following notations:

• Arg(·) denotes the principal determination of the argument of a complexnumber,

• for n ∈ {d1, d2}, Sn−1 is the n − 1-dimensional unit sphere,

• g is a linear form defined below,• the function s and the constants c(d1, d2) and J are defined by

s(x) = 1

2|x|(d2−2)/2, c(d1, d2) =

∫ 1

0rd1−1(1 − r2)(d2−2)/2 dr,

J =∫

Td2

1

h2(0, θ2)dθ2

• Q is the Hessian (d1+d2)×(d1+d2)-matrix of h at 0 that can be decomposed

as Q =(

Q1 R∗R Q2

).

The function g is defined as follows. We assume d2 � 3 and 1 + tI = 0.Let h2(θ1) = infθ2∈T

d2 h(θ1, θ2). In Section 5.1, we show that the function θ1 �→∫T

d2 (h(θ1, θ2) − h2(θ1))−1 dθ2 is real analytic in a neighborhood of 0. Using the

Taylor expansion of this function near 0, one obtains

1 + t

∫

Td2

1

h(θ1, θ2) − h2(θ1)dθ2 = tg(θ1) + O(|θ1|2).

This defines the linear form g uniquely. Then, g is defined by

g(θ) := (2π)d2√

Det(Q2)g((Q1 − RQ−12 R∗)−1/2θ1).

If the variables (θ1, θ2) separate in h, i.e., if h(θ1, θ2) = h1(θ1)+h2(θ2), the functiong is identically 0.


1.2. THE CASE OF A RANDOM SURFACE POTENTIAL

Let Vω be a random potential concentrated on the sublattice Zd1 × {0} ⊂ Z

d (d1 ischosen as in Section 0) of the form

Vω(γ1, γ2) ={

ωγ1 if γ2 = 0,0 if γ2 �= 0,

γ = (γ1, γ2) ∈ Zd1 × Z

d2 = Zd, (1.4)

and (ωγ1)γ1∈Zd1 is a family of i.i.d. bounded, nonconstant random variables.

Let ω± be respectively the maximum and minimum of the random variables(ωγ1)γ1∈Z

d1 , and let ω be its expectation.Finally, we define the random surfacic model by

Hω = H + Vω, (1.5)

and its IDSS by

(ϕ, d ns) = E(tr(�1[ϕ(Hω) − ϕ(H)]�1)).

Following Section 0, one regularizes ns into Ns as in (0.5).

Remark 1.1. An interesting case which can be brought back to a Hamiltonianof the form (1.5) with H and Vω as above is the following.

Consider �, a sub-lattice of Zd obtained in the following way � = G({0}×Z

d2)

where G is a matrix in GSLd(Z), the d-dimensional special linear group over Z,i.e. the multiplicative group of invertible matrices with coefficients in Z and unitdeterminant. One easily shows that the random operator

Hω(�) = −1

2� +

∑

γ∈�

ωγ �γ

(where �γ is the projector onto the vector δγ ∈ �2(Zd)) is unitarily equivalent toH + Vω where Vω is defined in (1.4) and h(θ) = h0(G

′ · θ); here, h0 is definedin (1.1) and G′ is the inverse of the transpose of G, i.e. G′ = tG−1.

DEFINITION 1.1. We say that E, an edge (or boundary) of the spectrum of Hω,is stable if it is an edge of the spectrum of H + tVω for all t ∈ [0, 1]. If an edge isnot stable, we call it a fluctuation edge.

Note that this definition is equivalent to the one given in the introduction withinthe context considered there.

As in the introduction, one has to distinguish between

(1) stable boundaries: at these boundaries, the IDSS is given by the IDSS of amodel operator computed from the random model and

(2) fluctuation boundaries: at these boundaries, one has standard Lifshitz tails.


To complete this section, let us give a very simple description of the spectrumof Hω. One has

PROPOSITION 1.2. Let Hω be defined as above. Then

σ(Hω) =⋃

t∈supp(P0)

σ (Ht).

Here and in the following P0 denotes the common distribution of the randomvariables (ωγ2)γ2 .

1.3. THE STABLE BOUNDARIES

The stable boundary we are studying is the lower boundary which we assumed tobe 0. Let us first give a criterion for the lower edge of the spectrum of H (whichwe assume to be equal to 0) to be a stable edge. We prove

PROPOSITION 1.3. Write h(θ) = h(θ1, θ2) where θ = (θ1, θ2), θ1 ∈ Td1 , θ2 ∈

Td2 . Then, 0 is a stable spectral edge if and only if t = ω− satisfies condition (1.3).

Proposition 1.3 is an immediate consequence of Propositions 1.1 and 1.2. Itgives the obvious fact that, if ω− � 0, then 0 is a stable edge. As we assumed thath has only nondegenerate minima, we see that if d2 = 1, 2 and ω− < 0, then 0 isnever a stable edge. Actually, it even need not be an edge of the spectrum of Hω.

Using the same notations as above, we prove

THEOREM 1.2. Assume (H0) and (H1) are verified. Assume, moreover, that 0 isa stable spectral edge for Hω. Then, one has

if ω > 0, then lim infE→0+

Ns(E)

Nωs (E)

� 1 and

if ω < 0, then lim supE→0+

Ns(E)

Nωs (E)

� 1,

(1.6)

where Nωs is the IDSS of the operator with constant surface potential ω, the com-

mon expectation value of the random variables (ωγ1)γ1 .

This result admits an immediate corollary

THEOREM 1.3. Assume (H0) and (H1) hold. Assume, moreover, that 0 is a stablespectral edge for Hω. Then,

• if d2 = 1:

Ns(E) ∼E→0+

Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)· E1+d1/2,


• if d2 = 2:

Ns(E) ∼E→0+

2Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)

E1+d1/2

| log E| .

Theorem 1.3 is an immediate consequence of Theorem 1.2 and the bound

Nω−s (E) � Ns(E) � Nω+

s (E).

As noted in the introduction, Theorem 1.2 is only useful when ω− = 0 (in whichcase ω > 0). Moreover, one obtains the analogue of (0.7) in the present case ford2 � 3.

The above results may lead to the belief that

Ns(E) ∼E→0

Nωs (E)

for all dimensions d2. Let us now explain why this result, if true, is not obtained fordimension d2 � 3. Therefore, we explain the heuristics behind the proof of Theo-rem 1.2; it is very similar to that of standard Lifshitz tails with one big differencewhen d2 � 3.

Restrict Hω to some large cube. One wants to estimate the IDSS for Hω; forthis restriction, this amounts to estimating the differences between the integrateddensity of states (the usual one) of the operator Hω and the integrated density ofstates of the operator Hω− (see Lemma 2.2). So we want to count the eigenvaluesof Hω below energy E, say, subtract the number of eigenvalues of Hω− belowenergy E, divide by the volume of the cube, and see how this behaves when E getssmall. Assume ϕ is a normalized eigenfunction associated to an eigenvalue of Hω

below E. Then, one has 〈(H +Vω)ϕ, ϕ〉 � E. Assume for a moment that Vω is non-negative. Then, we see that one must have both 〈Hϕ, ϕ〉 � E and 〈Vωϕ, ϕ〉 � E.The first of these conditions guarantees that ϕ is localized in momentum. So it hasto be extended in space. If one plugs this information into the second condition,one sees that 〈Vωϕ, ϕ〉 ∼ ω〈�2

0ϕ, ϕ〉 with a large probability. Therefore in the stateϕ, Hω roughly looks like H + ω�2

0. There is one problem with this reasoning: asVω only lives on a hypersurface, and as ϕ is flat, it only sees a very small part of ϕ;a simple calculation shows that ‖�2

0ϕ‖ ∼ Ed2/2; on the other hand, when one saysthat ϕ is roughly constant, one makes an error of size Eα (for some 0 < α < 1);hence, for dimension d2 � 3, this error is much larger than the term we want toestimate, namely, 〈Vωϕ, ϕ〉. In other words, because ϕ is very flat, we can modifyit on the hypersurface (e.g. localize the part of it living on the hypersurface) withalmost no change to the total energy of ϕ; hence, we cannot guarantee that ϕ is alsoflat on the hypersurface, which implies that 〈Vωϕ, ϕ〉 need not be close ω with alarge probability.


1.4. THE FLUCTUATION BOUNDARIES

In this section we assume that the infimum of � which we call E0 is (strictly)below inf(σ (H)), so that E0 is a fluctuation edge. In this case, we consider a‘reduced’ operator H which acts on �2(Zd1). In Fourier representation this operatoris multiplication by the function h given by:

h(θ1) =(∫

Td2

1

h(θ1, θ2) − E0dθ2

)−1

+ E0. (1.7)

We will reduce the proof of Lifshitz tails for Hω = H + Vω to a proof of Lifshitztails for the reduced operator Hω = H + Vω (where Vω is a diagonal matrixwith entries (ωγ1)γ1 ). To prove Lifshitz tail behavior for Hω we have to imposea condition on the behavior of h near its minimum. We either suppose:

(H2): the function h: Td1 → R admits a unique quadratic minimum.

or we assume the weaker hypothesis:

(H2′): the function h: Td → R is not constant.

Moreover, we always assume that the random variables ωγ1 defining the potential(0.1) are independent with a common distribution P0. We set ω− = inf(supp(P0))

and assume:

(H3): P0 is not concentrated in a single point and P0([ω−, ω− + ε)) � C εk forsome k > 0 and C > 0.

We will prove below:

THEOREM 1.4. If (H2) and (H3) are satisfied then

limE↘E0


= −d1

2.

We have an additional result when the surface has a low dimension:

THEOREM 1.5. Assume (H2′) and (H3) hold. If d1 = 1 then

limE↘E0


= − limE↘E0

ln(n(E − ω−))

ln(E − E0),

where n(E) is the integrated density of states for H .If d2 = 2, then

limE↘E0


= −α,

where the computation of α is explained below.


For the sake of simplicity, let us assume E0 = 0. The Lifshitz exponent α willdepend on the way h vanishes at S = {θ1|h = 0} and on the curvature of S.

To describe it precisely, we need to introduce some objects from analytic geom-etry (see [19] for more details). If E is a set contained in the closed first quadrantin R

2 then its exterior convex hull is the convex hull of the union of the rectanglesRxy = [x, ∞) × [y, ∞), where the union is taken over all (x, y) ∈ E .

Pick θ0 ∈ S and consider the Newton diagram of h at θ0, i.e.,

(1) Express h as a Taylor series at θ0, h(θ1, θ2) = ∑ij aij (θ

1 − θ10 )i(θ2 − θ2

0 )j ,θ = (θ1, θ2).

(2) Form the exterior convex hull of the points (i, j) with aij �= 0. This is a convexpolygon, called the Newton polygon.

(3) The boundary of the polygon is the Newton diagram.

The Newton decay exponent is then defined as follows. The Newton diagramconsists of certain line segments. Extend each to a complete line and intersect itwith the diagonal line θ1 = θ2. This gives a collection of points (ak, ak), one foreach boundary segment. Take the reciprocal of the largest ak and call this numberα(h, θ0); it is the Newton decay exponent. Define α(h, θ0) = min{α(h ◦ T0, θ0):T0(·) = θ0 + T (· − θ0), T ∈ SL(2, R)}.

Similarly, define α(h, θ) if θ is any other point in S, the zero set of h. Then, theLifshitz exponent α is defined by

α = minθ∈S

α(h, θ). (1.8)

The Lifshitz exponent α is positive as θ �→ α(h, θ) is a positive, lower semi-continuous function and S is compact (see [19]).

Remark 1.2. Let us return to the example given in Remark 1.1. In the Appendix,we check that (H.2′) holds in this case; so for d = d1+d2 = 3, Theorem 1.5 applies.

2. Approximating the IDSS

To approximate the IDSS, we use a method that has proved useful to approximatethe density of states of random Schrödinger operators, the periodic approximations.We shall show that the IDSS is well approximated by the suitably normalizeddensity of states of a well chosen periodic operator.

2.1. PERIODIC APPROXIMATIONS

Let (ωγ1)γ1∈Zd1 be a realization of the random variables defined above. Fix N ∈ N

∗.We define HN

ω , a periodic operator acting on �2(Zd) by

HNω = H + V N

ω = H +∑

γ1∈Zd12N+1

ωn

∑

β1∈(2N+1)Zd1

β2∈(2N+1)Zd2

|δγ1+β1 ⊗ δβ2〉〈δγ1+β1 ⊗ δβ2 |.


Here, Zd2N+1 = Z

d /(2N + 1)Zd , δl = (δjl)j∈Zd is a vector in the canonical basis

of �2(Zd ) where δjl is the Kronecker symbol and, d = d1 or d = d2, the choicebeing clear from the context. As usual, |u〉〈u| is the orthogonal projection on a unitvector u.

By definition, HNω is periodic with respect to the (nondegenerate) lattice

(2N + 1)Zd . We define the density of states denoted by nNω as usual for periodic

operators: for ϕ ∈ C∞0 (R),

(ϕ, dnNω ) =

∫

R

ϕ(x) dnNω (x) = lim

L→+∞1

(2L + 1)d

∑

γ∈Zd

|γ |�L

〈δγ , ϕ(HNω )δγ 〉.

This limit exists (see, e.g., [4, 23]). In a similar way, one can define the density ofstates of H ; we denote it by dn0. The operators (HN

ω )ω,N are uniformly bounded;hence, their spectra are contained in a fixed compact set, say C. This set alsocontains the spectrum of Hω and H . We prove

LEMMA 2.1. Pick U ⊂ R a relatively compact open set such that C ⊂ U. Thereexists C > 1 such that, for ϕ ∈ C∞

0 (R), for K ∈ N, K � 1, and N ∈ N∗, we have

|(ϕ, dns) − (2N + 1)d2E{(ϕ, [dnNω − dn0])}|

�(

CK

N

)K

supx∈U

0�J�K+d+2

∣∣∣∣dJ ϕ

dJ x(x)

∣∣∣∣. (2.1)

Proof. Fix ϕ ∈ C∞0 (R). As the spectra of the operators HN

ω are contained in U,we may restrict ourselves to ϕ supported in U which we do from now on. By thedefinition (0.2), one has

(ϕ, dns) = E

( ∑

γ2∈Zd2

〈δ0 ⊗ δγ2, [ϕ(Hω) − ϕ(H)]δ0 ⊗ δγ2〉)

= MN(ϕ) + RN(ϕ), (2.2)

where

MN(ϕ) = E

( ∑

γ2∈Zd2

|γ2|�N

〈δ0 ⊗ δγ2, [ϕ(Hω) − ϕ(H)]δ0 ⊗ δγ2〉)

,

RN(ϕ) = E

( ∑

γ2∈Zd2

|γ2|>N

〈δ0 ⊗ δγ2, [ϕ(Hω) − ϕ(H)]δ0 ⊗ δγ2〉)

.

Let us now show that

|RN(ϕ)| �(

CK

N

)K

supx∈U

0�J�K+d+2

∣∣∣∣dJ ϕ

dJ x(x)

∣∣∣∣. (2.3)


Therefore, we use some ideas from the proof of Lemma 1.1 in [17]. Helffer–Sjöstrand’s formula [10] reads

ϕ(Hω) = i

2π

∫

C

∂ϕ

∂z(z) · (z − Hω)−1 dz ∧ dz,

where ϕ is an almost analytic extension of ϕ (see [22]), i.e. a function satisfying

(1) for z ∈ R, ϕ(z) = ϕ(z);(2) supp(ϕ) ⊂ {z ∈ C; |Im(z)| < 1};(3) ϕ ∈ S({z ∈ C; |Im(z)| < 1});(4) the family of functions x �→ (∂ϕ/∂z)(x + iy) · |y|−n (for 0 < |y| < 1) is

bounded in S(R) for any n ∈ N; more precisely, there exists C > 1 such that,for all p, q, r ∈ N, there exists Cp,q > 0 such that

sup0<|y|�1

supx∈R

∣∣∣∣x

p ∂q

∂xq

(

|y|−r · ∂ϕ

∂z(x + iy)

)∣∣∣∣

� CrCp,q supq′�r+q+2

p′�p

supx∈R

∣∣∣∣x

p′ ∂q ′ϕ

∂xq ′ (x)

∣∣∣∣. (2.4)

As we are working with ϕ with compact support in U, its almost analyticextension can be taken to have support in (U + [−1, 1]) + i[−1, 1] (see, e.g., [6]).

We estimate E(|〈δ0 ⊗ δγ2, [ϕ(Hω) − ϕ(H)]δ0 ⊗ δγ2〉|) for |γ2| > N . Using thefact that the random variables (ωγ2)γ2 are bounded, we get

E(|〈δ0 ⊗ δγ2, [ϕ(Hω) − ϕ(H)]δ0 ⊗ δγ2〉|)

� 1

4πE

(∫

C

∣∣∣∣∂ϕ

∂z(z)

∣∣∣∣|〈δ0 ⊗ δγ2, ((z − HN

ω )−1 − (z − H)−1)δ0 ⊗ δγ2〉| dx dy

)

� C∑

γ1∈Zd1

∫

C

∣∣∣∣∂ϕ

∂z(z)

∣∣∣∣ · E(|〈δ0 ⊗ δγ2, (z − HN

ω )−1δγ1 ⊗ δ0〉|×

× |〈δγ1 ⊗ δ0, (z − H)−1δ0 ⊗ δγ2〉|) dx dy,

where z = x + iy.By a Combes–Thomas argument (see, e.g., [18]), we know that there exists

C > 1 such that, uniformly in (ωγ )γ , γ1 ∈ Zd1 and N � 1, we have, for Im(z) �= 0,

|〈δγ1 ⊗ δγ2, (z − HNω )−1δγ ′

1⊗ δγ ′

2〉| + |〈δγ1 ⊗ δγ2, (z − H)−1δγ ′

1⊗ δγ ′

2〉|

� C

|Im(z)|e−|Im(z)|(|γ1−γ ′1|+|γ2−γ ′

2|)/C. (2.5)

Hence, for some C > 1,

|RN(ϕ)| � C∑

γ1∈Zd1

∫

C

∣∣∣∣∂ϕ

∂z(z)

∣∣∣∣ · 1

|Im(z)|2 e−|Im(z)(|γ1|+|γ2|)|/C dx dy


� C

∫

C

∣∣∣∣∂ϕ

∂z(z)

∣∣∣∣

1

|Im(z)|d+2e−|Im(z)N |/C dx dy.

Taking into account the properties of almost analytic extensions (2.4), for someC > 1, for K � 1 and N � 1, we get

|RN(ϕ)| � CK+1∫

(U+[−1,1])+i[−1,1]|y|Ke−|yN |/C dx dy sup

x∈U0�J�K+d+2

∣∣∣∣dJ ϕ

dJ x(x)

∣∣∣∣

�(

CK

N

)K

supx∈U

0�J�K+d+2

∣∣∣∣dJ ϕ

dJ x(x)

∣∣∣∣.

This completes the proof of (2.3).We now compare MN(ϕ) to (2N + 1)d2E{(ϕ, [dnN

ω − dn0])}. Therefore, werewrite this last term as follows. Using the (2N + 1)Zd periodicity of HN

ω and H ,we get

∑

γ∈Zd

|γ |�N+L(2N+1)

〈δγ , ϕ(HNω )δγ 〉 = (2L + 1)d

∑

γ∈Zd

|γ |�N

〈δγ , ϕ(HNω )δγ 〉.

This gives

(2N + 1)d(ϕ, dnNω ) = E

( ∑

γ∈Zd

|γ |�N

〈δγ , ϕ(HNω )δγ 〉

)

. (2.6)

On the other hand, as the random variables (ωγ2)γ2 are i.i.d. and as H is Zd-

periodic, as in [18], one computes

E

( ∑

γ∈Zd

|γ |�N

〈δγ , ϕ(HNω )δγ 〉

)

= E

( ∑

γ1∈Zd1 , |γ1|�N

γ2∈Zd2 , |γ2|�N

〈δγ1 ⊗ δγ2, ϕ(HNω )δγ1 ⊗ δγ2〉

)

= (2N + 1)d1E

( ∑

γ2∈Zd2

|γ2|�N

〈δ0 ⊗ δγ2, ϕ(HNω )δ0 ⊗ δγ2〉

)

.

Combining this with (2.6), we get

(2N + 1)d2E[(ϕ, dnNω )] = E

( ∑

γ2∈Zd2

|γ2|�N

〈δ0 ⊗ δγ2, ϕ(HNω )δ0 ⊗ δγ2〉

)

.

Of course, such a formula also holds when HNω is replaced with H . In view of (0.2),

(2.3) and (2.2), to complete the proof of Lemma 2.1, we need only to prove

E

∣∣∣∣

∑

γ2∈Zd2

|γ2|�N

〈δ0 ⊗ δγ2, [ϕ(HNω ) − ϕ(Hω)]δ0 ⊗ δγ2〉

∣∣∣∣


�(

CK

N

)K

supx∈U

0�J�K+d+2

∣∣∣∣dJ ϕ

dJ x(x)

∣∣∣∣ (2.7)

for ϕ, K J and N as in Lemma 2.1.Proceeding as above, for γ2 ∈ Z

d2 , |γ2| � N , we estimate

|〈δ0 ⊗ δγ2, [ϕ(HNω ) − ϕ(Hω)]δ0 ⊗ δγ2〉|

� C

[ ∑

γ ′1∈Z

d1

γ ′2∈((2N+1)Zd2 )∗

+∑

γ ′1∈Z

d1 , |γ ′1|>N

γ ′2=0

] ∫

C

∣∣∣∣∂ϕ

∂z(z)

∣∣∣∣ dx dy ×

× E(|〈δ0 ⊗ δγ2, (z − HNω )−1δγ ′

1⊗ δγ ′

2〉| ×

× |〈δγ ′1⊗ δγ ′

2, (z − Hω)−1δ0 ⊗ δγ2〉|).

Here we used the fact that the operators Hω and HNω coincide in the cube

{|γ | � N}.As Hω satisfies the same Combes–Thomas estimate (2.5) as HN

ω , doing thesame computations as in the estimate for RN(ϕ), we obtain (2.7). This completesthe proof of Lemma 2.1. �

Obviously, one has an analogue of (2.1) for ns,norm, nts or nt

s,norm. One needsto replace HN

ω and H with their obvious counterparts, i.e. choose the randomvariables (ωγ2)γ2 to be the appropriate constant.

This enables us to prove

LEMMA 2.2. Fix I , a compact interval. Pick α > 0. There exists ν0 > 0 andρ > 0 such that, for γ ∈ [0, 1], E ∈ I , ν ∈ (0, ν0) and N � ν−ρ , one has

(2N + 1)d2E(NNnorm,ω(E − ν)) − e−ν−α

� Ns,norm(E) � (2N + 1)d2E(NNnorm,ω(E + ν)) + e−ν−α

, (2.8)

where NNnorm,ω = NN

ω −NNω− , and NN

ω (resp. NNω−) is the integrated density of states

of HNω (resp. HN

ω− , i.e. HNω where ωγ = ω− for all γ ).

Let us note here that one can prove a similar result for the approximation ofNt

s,norm by Nt,Ns,norm, hence, for that of Ns by NN

ω .

Proof. Pick ϕ a Gevrey class function of Gevrey exponent α > 1 (see [11]);assume, moreover, that ϕ has support in (−1, 1), that 0 � ϕ � 1 and that ϕ ≡ 1on (−1/2, 1/2]. Let E ∈ I and ν ∈ (0, 1), and set

ϕE,ν(·) = 1[0,E] ∗ ϕ

( ·ν

)

.


Then, by Lemma 2.1 and the Gevrey estimates on the derivatives of ϕ, there existC > 1 such that, for N � 1, k � 1 and 0 < ν < 1, we have

|(2N + 1)d2E((ϕE,ν, dNNnorm,ω)) − (ϕE,ν, dNs,norm)| � C(Nν)3

(Ck1+α

Nν

)k

.

(2.9)

We optimize the right-hand side of (2.9) in k. As a result there exists C > 1 suchthat, for N � 1 and 0 < ν < 1, we have

|(2N + 1)d2E((ϕE,ν, dNNnorm,ω)) − (ϕE,ν, dNs,norm)|

� C(N + ν−1)3 e−(Nν/C)1/(1+α)+C(Nν/C)−1/(1+α)

.

Now, there exist ν0 > 0 such that, for 0 < ν < ν0 and N � ν−1−η, we have

|(2N + 1)d2E((ϕE,ν, dNNnorm,ω)) − (ϕE,ν, dNs,norm)| � e−ν−η/(2α)

. (2.10)

By definition, ϕE,ν ≡ 1 on [0, E], and ϕE,ν has support in [−ν, E + ν] and isbounded by 1. As dNN

norm,ω and dNs,norm are positive measures, we have

E(NNnorm,ω(E)) � E((ϕE,ν, dNN

norm,ω)) � E(NNnorm,ω(E + ν)). (2.11)

Hence, by (2.10) and (2.11), we obtain

Ns,norm(E) � (ϕE,ν, dNs,norm)

= (2N + 1)d2E[(ϕE,ν, dNNnorm,ω)] +

+ [(ϕE,ν, dNs,norm) − (2N + 1)d2E((ϕE,ν, dNN

norm,ω))]

� (2N + 1)d2E(NNnorm,ω(E + ν)) + e−ν−η/(2α)

and

Ns,norm(E) � (ϕE−ν,ν, dNs,norm)

= (2N + 1)d2E[(ϕE−ν,ν, dNNnorm,ω)] +

+ [(ϕE−ν,ν, dNs,norm) − (2N + 1)d2E((ϕE−ν,ν, dNN

norm,ω))]

� (2N + 1)d2E(NNnorm,ω(E − ν)) − e−ν−η/(2α)

.

This completes the proof of Lemma 2.2. �

2.2. SOME FLOQUET THEORY

To analyze the spectrum of HNω , we use some Floquet theory that we develop now.

We identify Td with [−π, π ]d . Let us denote by F : L2([−π, π ]d) → �2(Zd)


the standard Fourier series transform. With θ = (θ1, θ2), we have, for u ∈L2([−π, π ]d),

(Hωu)(θ) = (F ∗HωF u)(θ) = h(θ)u(θ) +∑

β1∈Zd1

ωβ1(�β1u)(θ),

where

(�β1u)(θ) = 1

(2π)deiβ1θ1

∫

[−π,π]de−iβ1θ1u(θ) dθ.

Define the unitary equivalence

U : L2([−π, π ]d) → L2

([

− π

2N + 1,

π

2N + 1

]d)

⊗ �2(Zd2N+1),

u �→ (Uu)(θ) = (uγ (θ))γ∈Zd2N+1

,

where the (uγ (θ))γ∈Zd2N+1

are defined by

u(θ) =∑

γ∈Zd2N+1

eiγ θuγ (θ),

where the functions (θ �→ uγ (θ))γ∈Zd2N+1

are2π

2N + 1Z

d-periodic.

(2.12)

The functions (uγ )γ∈Zd2N+1

are computed easily; if the Fourier coefficients of u aredenoted by (uγ )γ∈Zd , then, one gets

uγ (θ) =∑

β∈Zd

uγ+(2N+1)β ei(2N+1)βθ . (2.13)

The operator UF ∗HNω F U ∗ acts on

L2

([

− π

2N + 1,

π

2N + 1

]d)

⊗ �2(Zd2N+1);

it is the multiplication by the matrix

MNω (θ) = HN(θ) + V N

ω , (2.14)

where

HN(θ) = ((hβ−β ′(θ)))(β,β ′)∈(Zd2N+1)

2

andV N

ω = ((ωβ1δβ1β′1δβ20δβ ′

20))(β1,β′1)∈(Z

d12N+1)

2,(β2,β′2)∈(Z

d22N+1)

2 .

(2.15)

Here, the functions (hγ )γ∈Zd2N+1

are the components of h decomposed according

to (2.12). The (2N + 1)d × (2N + 1)d-matrices HN(θ) and V Nω are nonnegative

matrices.This immediately tells us that the Floquet eigenvalues and eigenvectors of HN

ω

with Floquet quasi-momentum θ (i.e. the vectors, u = (uβ)β∈Zd ), solution to the


problem

HNω u = λu,

uβ+γ = e−iγ θuβ for β ∈ Zd, γ ∈ (2N + 1)Zd

are the eigenvalues and eigenvectors (once extended quasi-periodically) of the(2N + 1)d × (2N + 1)d matrix MN

ω (θ). For E ∈ R, one has

N Nω (E) =

∫ E

0dnN

ω (E)

=∫

[− π2N+1 , π

2N+1 ]d�{eigenvalues of MN

ω (θ) in [0, E]} dθ.

Considering H as (2N + 1)Zd-periodic on Zd , we see that the Floquet eigenvalues

of H (for the quasi-momentum θ ) are (h(θ+ 2πγ

2N+1))γ∈Zd2N+1

; the Floquet eigenvalue

h(θ + 2πγ

2N+1) is associated to the Floquet eigenvector uγ (θ), γ ∈ Zd2N+1 defined by

uγ (θ) = 1

(2N + 1)d/2(e−i(θ+ 2πγ

2N+1 )β)β∈Zd2N+1

.

In the sequel, the vectors in �2(Zd2N+1) are given by their components in the or-

thonormal basis (uγ (θ))γ∈Zd2N+1

. The vectors of the canonical basis denoted by(vl(θ))l∈Z

d2N+1

have the following components in this basis

vl(θ) = 1

(2N + 1)d/2(ei(θ+ 2πγ

2N+1 )l)γ∈Zd2N+1

.

We define the vectors (vl)l∈Zd2N+1

by

vl = e−ilθ vl(θ) = 1

(2N + 1)d/2(ei

2πγ l2N+1 )γ∈Z

d2N+1

.

3. The Proof of Theorem 1.2

To prove Theorem 1.2, we will use Lemma 2.2 and the Floquet theory developedin 2.2. We will start with

3.1. THE FLOQUET THEORY FOR CONSTANT SURFACE POTENTIAL

We consider the operator HNt = HN

ω where ω = (t)γ1∈Z

d12N+1

is the constant vector

and t �= 0. The matrix MNt (θ) defined by (2.14) for HN

t takes the form (2.14)where

V Nt = t ((δβ1β

′1δβ20δβ ′

20))(β1,β′1)∈(Z

d12N+1)

2, (β2,β′2)∈(Z

d22N+1)

2 . (3.1)

Our goal is to describe the eigenvalues and eigenfunctions of MNt (θ). As usual, we

write θ = (θ1, θ2). By definition, the operator HNt is Z

d1 × (2N + 1)Zd2 -periodic.It can be seen as acting on �2(Zd1, �2(Zd2)); as such, we can perform a Floquet


analysis in the θ1-variable as in Section 2.2 (in this case, just a discrete Fouriertransform in θ1) to obtain that HN

t is unitarily equivalent to the direct sum over θ1

in Td1 of the 2N + 1-periodic operator HN

t (θ1) acting on �2(Zd2) defined by thematrix

HNt (θ1) =

((

h(θ1;β2 − β ′2) + t

∑

γ2∈(2N+1)Zd2

δβ2γ2δβ ′2γ2

))

(β2,β′2)∈(Zd2 )2

.

Here h(θ1;β2) is the partial Fourier transform of h(θ1, θ2) in the θ2-variable.For each θ1, we now perform a Floquet reduction for HN

t (θ1) to obtain thatHN

t (θ1) is unitarily equivalent to the multiplication by the matrix

MNt (θ1, θ2) = ((h(θ1, θ2;β2 − β ′

2) + tδβ20δβ ′20))(β2,β

′2)∈(Z

d22N+1)

2 .

The matrix-valued function (θ1, θ2) �→ MNt (θ1, θ2) is 2πZ

d1-periodic in θ1 and2π

2N+1Zd2 -periodic in θ2. It is a rank one perturbation of the matrix MN

0 (θ1, θ2);

the eigenvalues of this matrix are the values h(θ1, θ2 + 2πγ22N+1). Let us for a while

order these values increasingly and call them (ENn (θ1, θ2, t)1�n�nN

) where nN �(2N + 1)d2 (we do not repeat the eigenvalues according to multiplicity). The stan-dard theory of rank one perturbations [24] yields

LEMMA 3.1. Assume t > 0. For 1 � n � nN , if ENn (θ1, θ2, 0) is an eigenvalue of

multiplicity k of MN0 (θ1, θ2), then

• either it is an eigenvalue of multiplicity k for MNt (θ1, θ2);

• or it is an eigenvalue of multiplicity k − 1 for MNt (θ1, θ2) and the interval

(ENn (θ1, θ2, 0), EN

n+1(θ1, θ2, 0)) contains exactly one simple eigenvalue; thiseigenvalue is given by the condition

t〈δ0, (E − MN0 (θ1, θ2))

−1δ0〉 = 1.

Here, we took the convention ENnN+1(θ1, θ2, 0) = +∞. One has a symmetric state-

ment for t < 0.

For 1 � n � nN , let (ϕNn,j (θ1, θ2, t))1�j�jn

denote orthonormalized eigenvec-tors associated to the eigenvalue EN

n (θ1, θ2, t) where jn denotes its multiplicity.In the sequel, it will be convenient to reindex the eigenvalues and eigenfunctions

of the matrix MNt (θ1, θ2) as (EN

γ2(θ1, θ2, t))γ2∈Z

d22N+1

and (ϕNγ2

(θ1, θ2, t))γ2∈Zd22N+1

.

Clearly, the functions (θ1, θ2) �→ ENγ2

(θ1, θ2, t) and (θ1, θ2) �→ ϕNγ2

(θ1, θ2, t) canbe chosen to be 2πZ

d1-periodic in θ1 and 2π2N+1Z

d2 -periodic in θ2.Let us now show the

LEMMA 3.2. The eigenvalues of MNt (θ) are the values {Eγ1,γ2(θ1, θ2, t); γ1 ∈

Zd12N+1, γ2 ∈ Z

d22N+1} where

Eγ1,γ2(θ1, θ2, t) = ENγ2

(

θ1 + 2πγ1

2N + 1, θ2, t

)

. (3.2)


A normalized eigenfunction associated to the eigenvalue ENγ2

(θ1 + 2πγ12N+1 , θ2, t) is

the vector

vγ1,γ2(θ1, θ2, t)

:= (2N + 1)−d1/2

(

e−iβ1(θ1+ 2πγ12N+1 )ϕN

γ2

(

θ1 + 2πγ1

2N + 1, θ2, t

))

β1∈Zd12N+1

, (3.3)

i.e. the vector of components

(2N + 1)−d1/2

(

e−iβ1(θ1+ 2πγ12N+1 )cβ2

γ2

(

θ1 + 2πγ1

2N + 1, θ2

))

β1∈Zd12N+1

β2∈Zd22N+1

(3.4)

if ϕNγ2

(θ1, θ2, t) has components (cβ2γ2

(θ1, θ2))β2∈Zd22N+1

.

The vectors (vγ1,γ2(θ1, θ2, t))γ1∈Zd12N+1,γ2∈Z

d22N+1

form an orthonormal basis of

�2(Zd12N+1 × Z

d22N+1).

Proof. Orthonormality is easily checked using the fact that the vectors(ϕN

γ2(θ1, θ2, t))γ2∈Z

d22N+1

form an orthonormal basis.

Let us now check that vγ1,γ2(θ1, θ2, t) satisfies the eigenvalue equation for MNt (θ)

and Eγ1,γ2(θ1, θ2, t) given in (3.2). Therefore, first note that the matrix MNt (θ) is

nothing but the multiplication operator by the matrix-valued function MNt (θ1) to

which one has applied the Floquet reduction of in the θ1-variable. Hence, by (2.13),the matrix elements of MN

t (θ) given by (2.15) satisfy, for β1 ∈ Zd12N+1,

MNt

(

θ1 + 2πγ1

2N + 1

)

e−iβ1(θ1+ 2πγ12N+1 ) =

∑

β ′1∈Z

d12N+1

mβ1−β ′1(θ1)e

−iβ ′1(θ1+ 2πγ1

2N+1 ). (3.5)

Both sides in this equality are matrices acting on �2(Zd22N+1), the matrices mβ1−β ′

1(θ)

being defined as

mβ1−β ′1(θ) = ((hβ1−β ′

1,β2−β ′2(θ)))

(β2,β′2)∈(Z

d22N+1)

2 .

If we now apply both sides of Equation (3.5) to the vector ϕNγ2

(θ1 + 2πγ12N+1 , t), we

obtain, for β1 ∈ Zd12N+1,

∑

β ′1∈Z

d12N+1

hβ1−β ′1(θ1)e

−iβ ′1(θ1+ 2πγ1

2N+1 )ϕNγ2

(

θ1 + 2πγ1

2N + 1, θ2, t

)

= MNt

(

θ1 + 2πγ1

2N + 1

)


γ2

(

θ1 + 2πγ1

2N + 1, t

)

= ENγ2

(

θ1 + 2πγ1

2N + 1, θ2, t

)


γ2

(

θ1 + 2πγ1

2N + 1, θ2, t

)

.


Rewritten this is

MNt (θ)vγ1,γ2(θ1, θ2, t) = EN

γ2

(

θ1 + 2πγ1

2N + 1, θ2, t

)

vγ1,γ2(θ1, θ2, t)

and completes the proof of Lemma 3.2. �In the course of the proof of Theorem 1.2, we will use the

LEMMA 3.3. Fix t such that t > 0 if d2 = 1, 2 and 1 + tI∞ > 0 if d2 � 3. Then,for ρ > 2, there exists C > 0 such that, for N � E−ρ and E sufficiently small, theeigenvalues of MN

t satisfy

Eγ1,γ2(θ1, θ2, t) � E �⇒(

1 + |γ1|2N + 1

)2

� CE. (3.6)

Proof. When t is positive, (3.6) is clear by Lemmas 3.2 and 3.1, that is, by theintertwining of the eigenvalues of MN

0 (θ) and MNt (θ), and as the eigenvalues of

MN0 (θ) are the values h(θ1 + 2πγ1

2N+1 , θ2 + 2πγ22N+1) which satisfy (3.6) as h(θ) � C|θ |2.

Assume now that d2 � 3 and t satisfies 1 + tI∞ > 0. To complete the proof ofLemma 3.3, by Lemma 3.2, it is then enough to prove that, there exists C > 0 suchthat

|θ1|2 > CE �⇒ ∀γ2, ENγ2

(θ1, θ2, t) > E.

By the intertwining properties and the properties of h, this is clear except for thelowest of the (EN

γ2)γ2 . Assume now that |θ1|2 � E. Then, by our assumptions on the

behavior of h near its minimum, for some C > 0, one has that (θ1, e) �→ I (θ1, e)

is real analytic in {|θ1|2 � E}× {|e| � E/C}. Hence, using a standard estimate forRiemann sums, we get that, for |θ1|2 � E and |e| � E/C,

1 + t〈δ0, (MN0 (θ1) − e)−1δ0〉 = 1 + tI (θ1, e) + O(E−2Eρ).

So that, as 1 + tI∞ > 0, for E sufficiently small, the equation 1 + t〈δ0, (MN0 (θ1)−

e)−1δ0〉 = 0 has no solution for |θ1|2 � E and |e| � E/C. By the above discussion,this implies that, all the EN

γ2(θ1, θ2, t) lie above E/C. This completes the proof of

Lemma 3.3. �

3.2. THE PROOF OF THEOREM 1.2

We now have all the tools necessary to prove Theorem 1.2. Notice that, as ω > ω−,as 1 + ω−I∞ � 0, we know that 1 + ωI∞ > 0. So the asymptotics for Nω

s (E) aregiven by

Nωs (E) ∼

E→0+ C(ω) · f (E).


The precise value of the constant C(ω) and of the function f (E) are given inTheorem 1.1. The constant C(ω) is a continuous function of ω; and, for any c ∈ R,the function f (E) satisfies f (E + cE2) ∼ f (E) when E → 0; moreover, f is atmost polynomially small in E. All these facts will be useful.

We start with the proof of (1.6). We will use Lemma 2.2. As above, fix N largebut not too large, say N ∼ E−ρ for some large ρ. Fix δ > 0 small. Considerthe matrix MN

ω+δ(θ) obtained by the Floquet reduction of HN + (ω + δ)�20. Let

HNδ (E, θ) be the spectral space of MN

ω+δ(θ) associated the eigenvalues less that E.

LEMMA 3.4. Fix δ > 0, ρ > 2 and α ∈ (0, 1/2). For N ∼ E−ρ and E sufficientlysmall, with a probability at least 1 − e−E−α

, for all θ and all ϕ ∈ HNδ (E, θ), one

has

〈MNω (θ)ϕ, ϕ〉 � E‖ϕ‖2.

This lemma immediately implies the desired lower bound. Indeed, it impliesthat, for N ∼ E−ρ , with a probability at least 1 − e−E−α

, one has

NNω+δ(E) =

∫

[− π2N+1 , π


ω+δ(θ) in [0, E]} dθ

�∫

[− π2N+1 , π


ω (θ) in [0, E]} dθ

= NNω (E).

Taking the expectation of both side, and using (2.8) for NNω+δ and NN

ω (and the factthat the number of eigenvalues of Mn

ω(θ) and MNω+δ(θ) are bounded by (2N +1)d),

we obtain

Nω+δs (E − E2) − CEdρ e−E−α � Ns(E).

Considering the remarks made above, we obtain

C(ω) � lim infE→0+

Ns(E)

f (E).

As C(ω) has the same sign as ω, this completes the proof of (1.6).

Proof of Lemma 3.4. Pick E small and ϕ ∈ HNδ (E, θ). Then, by Lemma 3.3,

ϕ can be expanded as

ϕ =∑

|γ1|�CE1/2N

γ2∈Zd22N+1

aγ1,γ2vγ1,γ2(θ, ω + δ),

where the vectors (vγ (θ))γ are given by (3.3) and (3.4). Using these equations, wecompute

〈V Nω ϕ, ϕ〉 =

∑

β1∈Zd12N+1

ωβ1 |Aβ1 |2, (3.7)


where

Aβ1 = 1

(2N + 1)d1/2

∑

|γ1|�CE1/2N

ei2πβ1 ·γ1

2N+1 cγ1 and

cγ1 =∑

γ2∈Zd22N+1

aγ1,γ2

⟨

δ0, ϕNγ2

(

θ1 + 2πγ1

2N + 1, θ2, ω + δ

)⟩

.(3.8)

So the vector (Aβ1)β1 is the discrete Fourier transform of the vector c = (cγ1)γ1

supported in a ball of radius CE1/2N . To estimate this Fourier transform, we usedthe following result

LEMMA 3.5 [18]. Assume N , L, K , K ′ L′ are positive integers such that

• 2N + 1 = (2K + 1)(2L + 1) = (2K ′ + 1)(2L′ + 1),• K < K ′ and L′ < L.

Pick a = (an)n∈Zd2N+1

∈ �2(Zd2N+1) such that,

for |n| > K, an = 0.

Then, there exists a ∈ �2(Zd2N+1) such that

(1) ‖a − a‖�2(Zd2N+1)

� CK,K ′‖a‖�2(Zd2N+1)

where CK,K ′ �K/K ′→0 K/K ′;(2) write a = (aj )j∈Z

d2L+1

; for l′ ∈ Zd2L′+1 and k′ ∈ Z

d2K ′+1, we have

∑

j∈Zd2L+1

aj ei2πj ·(l′+k′(2L′+1))

2N+1 =∑

j∈Zd2L+1

aj ei2πj ·k′2K ′+1 ;

(3) ‖a‖�2(Zd2N+1)

= ‖a‖�2(Zd2N+1)

.

This lemma is a quantitative version of the Uncertainty Principle; it says that,if a vector is localized in a small neighborhood of 0 (here, of size K/N ), up to asmall error δ, its Fourier transform is constant over cube of size N/(δK).

To apply Lemma 3.5, we pick N such that (2N +1) = (2K +1)(2L′+1)(2M+1) where K � CE1/2N ; this is possible as N ∼ E−ρ with ρ large; we pick forexample, L′ ∼ CE−(1−ν)/2 and M ∼ CE−ν/2 (for some fixed 0 < ν < 1). So2K ′ + 1 = (2K + 1)(2M + 1) and 2L + 1 = (2L′ + 1)(2M + 1).

We apply Lemma 3.5 to the vector c = (cγ1)γ1 defined in (3.8); by Lemma 3.5,there exists c = (cγ1)γ1 so that, if we set

Aβ1 = 1

(2N + 1)d1/2

∑

γ1∈Zd12N+1

ei2πβ1 ·γ1

2N+1 cγ1

then, for γ ′1 ∈ Z

d2L′+1 and β ′

1 ∈ Zd2K ′+1, we have

Aγ ′1+β ′

1(2L′+1) = Aβ ′1(2L′+1). (3.9)


Fix η > 0 small to be chosen later. We replace A by A in (3.7) and use theboundedness of the random variables to obtain

〈V Nω ϕ, ϕ〉 � (1 + η)

∑

β1∈Zd12N+1

ωβ1 |Aβ1 |2 + C

η‖A − A‖2.

Using (3.9) and points (1) and (3) of Lemma 3.5, we get that

〈V Nω ϕ, ϕ〉�

∑

β ′1∈Z

d12K ′+1

[C

ηEν/2 + 1

(2L′ + 1)d1

( ∑

γ ′1∈Z

d12L′+1

(1 + η)ωγ ′1+β ′

1(2L′+1)

)]

×

× (2L′ + 1)d1 |Aβ ′1(2L′ + 1)|2

Pick η such that η · ω+ < δ/4 and E sufficiently small that CEν/2 < δη/4. Wethen obtain

〈V Nω ϕ, ϕ〉 �

∑

β ′1∈Z

d12K ′+1

[

δ/2 + 1

(2L′ + 1)d1

( ∑

γ ′1∈Z

d12L′+1

ωγ ′1+β ′

1(2L′+1)

)]

×

× (2L′ + 1)d1 |Aβ ′1(2L′ + 1)|2 (3.10)

Now, if ω satisfies

∀β ′1 ∈ Z

d12K ′+1,

1

(2L′ + 1)d1

∑

γ ′1∈Z

d12L′+1

ωγ ′1+β ′

1(2L′+1) � ω + δ

2

then, (3.10) gives

〈V Nω ϕ, ϕ〉 � (ω + δ)

∑

β ′1∈Z

d12K ′+1

(2L′ + 1)d1 |Aβ ′1(2L′ + 1)|2 = 〈V N

ω+δϕ, ϕ〉,

where V Nt is defined in (3.1). Here, we have used the points (2) and (3) of

Lemma 3.5, and Definition (3.8) of the vector c = (cγ1)γ1 .To sum up, we have proved

LEMMA 3.6. Pick 0 < ν < 1. Pick N as described above. For E sufficientlysmall, the probability that, for all θ and all ϕ ∈ HN

δ (E, θ), one has

〈MNω (θ)ϕ, ϕ〉 � E‖ϕ‖2

is larger than the probability of the set{

ω; ∀β ′1 ∈ Z

d12K ′+1,

1

(2L′ + 1)d1

∑

γ ′1∈Z

d12L′+1

ωγ ′1+β ′

1(2L′+1) � ω + δ/2

}

.


The probability of this event is estimated by the usual large deviation estimates(see, e.g., [5, 7]). This completes the proof of Lemma 3.4. �

4. The Fluctuating Edges

In this section, we investigate the behavior of the density of surface states Ns(E)

at the bottom E0 of the spectrum of Hω in the case when E0 < inf σ(H) = 0. Aswe saw in Section 0.1, this is always the case for dimension d2 = 1 or d2 = 2and it holds in arbitrary dimensions if the support of common distribution P0 of theωγ1 has a sufficiently negative part. Thus, we are looking at a fluctuation edge asdescribed in Section 0.2. Due to the symmetry of the problem we may, of course,consider the top of the spectrum in an analogous way.

4.1. A REDUCED HAMILTONIAN

In the present situation it is convenient to think of the Hilbert space �2(Zd1+d2) as adirect some of �2(Zd1 × {0}) =: Hb and �2(Zd1+d2 \ Z

d1 × {0}) =: Hs , the indicesreferring to ‘bulk’ and ‘surface’ respectively (see [13] whose notations we follow).According to the decomposition H = HS ⊕ Hb we can write any operator A onH as a matrix

A =[

Ass Asb

Abs Abb

]

,

where Ass and Abb act on Hs and Hb respectively and Asb: Hb → Hs, Abs : Hs →Hb ‘connect’ the two Hilbert spaces Hs and Hb. The bounded operator A issymmetric if A∗

ss = Ass , A∗bb = Abb and A∗

sb = Abs . In the case of our randomHamiltonian Hω we have: (Hω)ss = (H0)ss + Vω while (Hω)bb = (H0)bb andHsb as well as Hbs are independent of the randomness. Moreover, by assumption,(Hω)bb � 0, while inf σ(Hω) < 0. Consequently, the operator ((H0)bb − E1bb)

−1

exists for all E < 0 and the operator

Gs(E) := (H0)ss + Vω − Hsb((H0)bb − E1bb)−1Hbs − E1ss

the so called resonance function is well defined. The operator Gs(E) is a sort of areduced Hamiltonian. Its inverse plays the role of a resolvent. It is not hard to showthat the set R(Hω) = {E ∈] − ∞, 0[; 0 ∈ σ(Gs(E))} (the resonant spectrum)agrees with the negative part of σ(Hω). See Prop. 1.2 in [13] for details. For laterreference, we state this as a lemma:

LEMMA 4.1. For E < 0, E is an eigenvalue of Hω if and only if 0 is an eigenvalueof Gs(E). Moreover the multiplicities agree.


In fact a little linear algebra proves that, for block matrices, we have(

A B

C D

)−1

=(

(A − BD−1C)−1 −A−1B(D − CA−1B)−1

−D−1C(A + BD−1C)−1 (D − CA−1B)−1

)

(4.1)when all the terms make sense.

We denote by N(A, E) the number of eigenvalues (counted according to multi-plicity) of the operator A below E. For �L = [−L, L]d we set (Hω,L)ij = (Hω)ij

if i, j ∈ �L and (Hω,L)ij = 0 otherwise. For energies E below zero the integrateddensity of surface states of Hω is given by

Ns(E) = limL→∞

1

(2L + 1)d1N(Hω,L, E).

Defining

GLs (E) = (Hω,L)ss − (HL)sb((HL)bb − E1bb)

−1(HL)bs − E1ss .

We have, as above, that E < 0 is an eigenvalue of Hω,L if and only if 0 is aneigenvalue of GL

s (E).In the following, we will express the density of surface states Ns(E) (for E < 0)

in terms of the operators GLs (E).

LEMMA 4.2. The eigenvalues ρn(E) of GLs (E) are continuous and decreasing

functions of E (for E < 0).Proof. Continuity is obvious from the explicit form of the entries of the (finite-

dimensional) matrix GLS (E). Let 0 > E2 > E1, then

GLs (E1) − GL

s (E2)

= −Hsb((Hbb − E1)−1 − (Hbb − E2)

−1)Hbs − (E1 − E2)

= (E2 − E1)Hsb((Hbb − E1)−1(Hbb − E2)

−1)Hbs + (E2 − E1).

Since E1, E2 < 0 the operator (Hbb−E1)−1(Hbb−E2)

−1 is positive, so the operatorGL

s (E1) − GLs (E2) is positive as well. �

PROPOSITION 4.1. For E < 0:

N(Hω,L, E) = N(GLs (E), 0).

Proof. For E sufficiently negative, Gs(E) is a positive operator. Let us nowincrease E (toward E = 0). Then, E is an eigenvalue of Hω,L if one of theeigenvalues of GL

s (E) passes through zero and becomes negative. �It follows from this proposition that (for E < 0)

Ns(E) = limL→∞ N(GL

s (E), 0)


GLs (E) depends on E in a rather complicated way through the resonance func-

tion. We will therefore approximate GLs (E) by an operator with much simpler

dependence on E in the following way: let E0 = inf σ(Hω) then we set:

GLs (E) = (Hω,L)ss − (HL)sb((HL)bb − E0)

−1(HL)bs − E.

This operator should give a good estimate for the eigenvalues of Hω near E0, infact:

LEMMA 4.3. For E0 < E < 0:

N(GLs (E), 0) � N(GL

s (E), 0).

Proof.

GLs (E) − GL

s (E) = (E − E0)Hsb((Hbb − E)−1(Hbb − E0)−1)Hbs.

So

GLs (E) � GL

s (E). �For a bound in the other direction we observe that:

LEMMA 4.4. For E0 � E � E1 < 0 we have

GLs (E) − GL

s (E) � C(E − E0).

Remark. The constant C in the above estimate depends on E0 and E1.

Proof of Lemma 4.4.

GLs (E) − GL

s (E) = (E − E0)Hsb((Hbb − E)−1(Hbb − E0)−1)Hbs

� (E − E0)Hsb((Hbb − E1)−1(Hbb − E0)

−1)Hbs

� C(E − E0).

Here, we used that

(Hbb − E)−1 � (Hbb − E1)−1. �

Summarizing, we have obtained

PROPOSITION 4.2. There is a constant C, such that for E0 � E � E0/2 < 0

N(GLs (E), 0) � N(Hω, E) � N(GL

s (E) − C(E − E0), 0).


The advantage of having GLs (E) rather than GL

s (E) lies in the fact that GLs (E)

depends linearly on E, in fact:

GLs (E) = Hss − Hsb(Hbb − E0)

−1Hbs + Vω − E

= H + Vω − E,

where H is the operator

H = Hss − Hsb(Hbb − E0)−1Hbs.

This operator is of a similar form as the Hamiltonian H , however it acts on �2(Zd1),i.e. on the surface only where the random potential Vω lives. The price to pay is thecomplicated looking “bulk term” Hsb(Hbb − E0)

−1Hbs .Nevertheless, H is still a Toeplitz operator and it is not too hard to compute its

symbol, i.e. its Fourier representation.In fact, a look at formula (4.1) shows that

H = [((H − E0)−1)ss]−1 + E0. (4.2)

Consequently the symbol of H is given by

h(θ1) =(∫

1

h(θ1, θ2) − E0dθ2

)−1

+ E0.

We summarize these results in a theorem:

THEOREM 4.1. Let Hω = H +Vω as in (1.5) satisfying assumption (H1). Assumemoreover, that E0 = inf σ(Hω) � 0. Define Hω = H + Vω as in (4.2) and letNs(Hω, E) be the integrated density of surface states of Hω and N(Hω, E) theintegrated density of states for Hω. Then

limE↘E0

ln | ln Ns(Hω, E)|ln(E − E0)

= limE↘E0

ln | ln N(Hω, E)|ln(E − E0)

,

where the equality should be interpreted in the following way: if one of the sidesexists so does the other one and they agree.

In other words, the Lifshitz exponent for the density of surface states of Hω andand the Lifshitz exponent for the density of states for Hω agree.

4.2. LIFSHITZ TAILS

In this section we investigate the integrated density of surface states Ns(E) forthe operator Hω = H + Vω acting on �2(Zd1 × Z

d2). We assume throughout thatE0 = inf σ(Hω) is (strictly) negative and E < 0.

By the previous section the investigation of Ns(E) for E near E0 can be reducedto estimates for the integrated density of states N(E) of the operator Hω = H + Vω


which acts on �2(Zd1). Hence the problem of surface Lifshitz tails boils down toordinary Lifshitz tails in a lower dimensional configuration space. However the(free) operator is somewhat more complicated, in fact in Fourier representation itis multiplication by

h(θ1) =(∫

1

h(θ1, θ2) − E0dθ2

)−1

+ E0.

We remind the reader that Vω(γ1) = ωγ1 for γ1 ∈ Zd1 and (ωγ1)γ1∈Z

d1 is a family ofindependent random variables with a common distribution P0.

Throughout this section we assume that supp(P0) is a compact set. Moreover,if we set ω− = inf(supp(P0)) we suppose that P0([ω−, ω− + ε) � Cεk) for somek > 0, C > 0.

THEOREM 4.2. If h has a unique quadratic minimum then

limE↘E0


= −d1

2.

Proof. The theorem follows from [16, 19] and the considerations above. �For dimensions d1 = 1 and d1 = 2 we have the following result:

THEOREM 4.3. Assume that h is not constant. If d1 = 1, then

limE↘E0


= − limE↘E0

ln(n(E − ω−))

ln(E − E0),

where n(E) is the integrated density of states for H .If d2 = 2, then

limE↘E0


= −α,

where α is defined in (1.8).

Note that n(E) ∼ (E − E0)ρ for some ρ > 0. See [16, 19] for details.

To conclude this section we consider some examples that fulfill the assumptionsof the previous theorems. Let us first assume that H is separable, i.e. that

h(θ1, θ2) = h1(θ1) + h2(θ2).

This is satisfied for example by the discrete Laplacian where h is equal to h0 givenin (1.1). The function h has a unique quadratic minimum if and only if both h1

and h2 have unique quadratic minima (which we may assume to be attained atθ1 = θ2 = 0).


We will show in the following that the function

h(θ1) =(∫

1

h1(θ1) + h2(θ2) − E0dθ2

)−1

+ E0

has a unique quadratic minimum in this case as well. Differentiating the function

ρ(θ1) =∫

1

h1(θ1) + h2(θ2) − E0dθ2

we obtain

∇ρ(θ1) = −∫ ∇h1(θ1)

(h1(θ1) + h2(θ2) − E0)2dθ2

so the (possible) maximum of ρ is at θ1 = 0.The second derivative at θ1 = 0 is given by

∇∇ρ(0) = −∇∇h1(0)

∫1

(h1(0) + h2(θ2) − E0)2dθ2

which obviously gives a negative definite Hessian.We remark that no assumptions on h2 were needed; in fact, the above arguments

work for h2 = const as well.The same reasoning also shows that h is not constant as long as h1 is not

constant.So we have proved:

THEOREM 4.4. Suppose h(θ1, θ2) = h1(θ1) + h2(θ2) then:

(1) If h1 has a unique quadratic minimum, then

limE↘E0


= −d1

2.

(2) If d1 = 1 and h1 is not constant then

limE↘E0


= − limE↘E0

ln(n(E − ω−))

ln(E − E0),

where n(E) is the integrated density of states for H .(3) If d2 = 2, then

limE↘E0


= −α,

where α is defined in (1.8).


5. The Density of Surface States for a Constant Surfacic Potential

In this section, we prove some useful results on the density of surface states fora constant surface potential. In some cases, this density may even be computedexplicitly (see, e.g., [2]).

The model we consider is the model introduced in Proposition 1.2 namely Ht =H + t1 ⊗ �2

0 where H is chosen as in Section 1 and t is a real coupling constant.The proof of all the results we now state is based on rank one perturbation theory(see, e.g., [24]). The main formula that we will use is the following: for z /∈ R, onehas

(Ht − z)−1 − (H − z)−1 = −t (H − z)−1 1 ⊗ �20

1 + tI (z) ⊗ 1(H − z)−1, (5.1)

where I (z) is the operator acting on �2(Zd1) that, in Fourier representation, is themultiplication by the function I (θ1, z) defined in (1.2).

Formula (5.1) is easily proved if one makes a partial Fourier transform in the(γ1, θ1) variable of H and Ht . If one does so, one obtains a direct integral repre-sentation for both H and Ht namely

H =∫

Td1

H(θ1) dθ1 and Ht =∫

Td1

Ht(θ1) dθ1,

where H(θ1) and Ht(θ1) (both acting on �2(Zd2)) differ only by a rank one operator,namely,

Ht(θ1) − H(θ1) = t�20.

Formulae (5.1) and (1.2) then follow immediately from the well known resolventformula for rank one perturbations that can be found, e.g., in [24].

Proposition 1.3 follows immediately from Proposition 1.2 and formulae (5.1)and (1.2). Indeed, by formula (5.1) and the special form of the operator I (z), z is apoint in σ(Ht) \ σ(H) if and only if, for some θ1, one has

1 + tI (θ1, z) = 0.

If we pick z ∈ R below 0 (recall that 0 = inf(σ (H)) = inf(h(Rd)), we see thatz ∈ σ(Ht) if and only if tI (θ1, z) = −1 for some θ1. As, for z < 0, I (θ1, z) is anegative decreasing function of z that tends to 0 when z → −∞, we see that thiscan happen if an only if tI (θ1, 0) < −1 for some θ1. This is the first statement ofProposition 1.1. Indeed, the function θ1 �→ tI (θ1, 0) is continuous of T

d1 except,possibly, at the points where h assumes its minimum, and it takes its minimal valueexactly at one of those points.

As, for the second statement, let I (z) := maxθ1∈Td1 I (θ1, z) and consider the

function f : z �→ 1 + tI (z). This function is clearly continuous and strictly de-creasing on ]−∞, 0[ and by assumption, it is negative near 0 (as 1 + tI∞ < 0)and f (z) → 1 as z → −∞. So, the function f admits a unique zero that we


denote by E0. The analysis given above immediately shows that E0 is the infimumof Ht : as θ1 �→ I (θ1, z) is continuous on T

d1 that is compact, for some θ1, one has1+ tI (θ1, E0) = 0. Hence E0 belongs to σ(Ht); on the other hand, for E < E0, forany θ1, one has 1+ tI (θ1, E) � 1+ tI (E) > 0, hence, E /∈ σ(Ht). This completesthe proof of Proposition 1.1.

5.1. ASYMPTOTICS OF THE DENSITY OF SURFACE STATES

The starting point for this computation is again formula (5.1). This enables us toget a very simple formula for the Stieltjes–Hilbert transform of the density of sur-face states nt

s for the pair (Ht , H). Using the Fourier representation and Parseval’sformula, one computes

tr(�1[(Ht − z)−1 − (H − z)−1]�1)

=∑

γ2∈Zd2

∫

Td1

−t

1 + tI (θ1, z)

∫

Td2

eiγ2θ2 dθ2

h(θ1, θ2) − z

∫

Td2

eiγ2θ2 dθ2

h(θ1, θ2) − zdθ1

=∑

γ2∈Zd2

∫

Td1

−t

1 + tI (θ1, z)

∫

Td2

e−iγ2θ2 dθ2

h(θ1, θ2) − z

∫

Td2

eiγ2θ2 dθ2

h(θ1, θ2) − zdθ1

=∫

Td1

−t

1 + tI (θ1, z)

∫

Td2

dθ2

(h(θ1, θ2) − z)2dθ1.

One then notices that∫

Td1

−t

1 + tI (θ1, z)

∫

Td2

dθ2

(h(θ1, θ2) − z)2dθ1

= − d

dz

∫

Td1

log(1 + tI (θ1, z)) dθ1.

Here, and in the sequel, log denotes the principal determination of the logarithm.This immediately yields that the Stieltjes–Hilbert transform of Nt

s is given by⟨

1

· − z, dNt

s

⟩

=∫

Td1

log(1 + tI (θ1, z)) dθ1,

where I is defined by (1.2).It is well known that one can invert the Stieltjes–Hilbert transform to recover

the signed measure dNts (see, e.g., the appendix of [23]). By the Stieltjes–Perron

inversion formula, one has∫ E

0dNt

s (e) = limε→0+

1

2iπ

∫ E

0

(⟨1

· − e − iε, dNt

s

⟩

−⟨

1

· − e + iε, dNt

s

⟩)

de

= limε→0+

1

2iπ

∫ E

0

∫

Td1

[log(1 + tI (θ1, e + iε)) −− log(1 + tI (θ1, e − iε))] dθ1 de. (5.2)


Notice that, for e real,

Im(1 + tI (θ1, e + iε)) = tε

∫

Td2

1

(h(θ1, θ2) − e)2 + ε2dθ2;

hence, this imaginary part keeps a fixed sign. So, for θ1 ∈ Td1 , one has

log(1 + tI (θ1, e + iε)) − log(1 + tI (θ1, e − iε))

= log

(1 + tI (θ1, e + iε)

1 + tI (θ1, e − iε)

)

.

For e ∈ R, one has |1 + tI (θ1, e + iε)| = |1 + tI (θ1, e − iε)|. As moreover theimaginary part of 1 + tI (θ1, e + iε) keeps a fixed sign, one has

| log(1 + tI (θ1, e + iε)) − log(1 + tI (θ1, e − iε))| � 2π.

As Td1 and [0, E] are compact, one can apply Lebesgue’s dominated convergence

Theorem to (5.2) and thus obtain∫ E

0dNt

s (e) =∫ E

0

∫

Td1

f (θ1, e) dθ1 de, (5.3)

where

f (θ1, e) = limε→0+

1

2iπlog

(1 + tI (θ1, e + iε)

1 + tI (θ1, e − iε)

)

= limε→0+

1

πArg(1 + tI (θ1, e + iε)), (5.4)

where Arg is the principal determination of the argument of a complex number.Notice here that this formula is the analogue of the well-known Birman–Kreınformula (see, e.g., [1, 25]) for surface perturbations.

We will now compute the asymptotics of f (θ1, e) for e small. First, let us noticethat we need only to compute these for θ1 small, i.e. close to 0. Indeed, we have as-sumed that h takes its minimum only at 0. Therefore, as T

d , is compact, if |θ1| � δ,we know that, for some δ′ > 0, for all θ2, one has h(θ1, θ2) � δ′. Hence, if |θ1| � δ,the function I (θ1, z) is analytic in a neighborhood of 0, so that f (θ1, e) = 0 for e

sufficiently small (independent of θ1). So, we now assume that |θ1| < δ for someδ > 0 to be chosen later on.

We now study I (θ1, z) for |z| small. Pick χ a smooth cut-off function in θ2, i.e.such that χ(θ2) = 1 if |θ2| � δχ and χ(θ2) = 0 if |θ2| � 2δχ . Write

I (θ1, z) =∫

Td2

χ(θ2)

h(θ1, θ2) − zdθ2 +

∫

Td2

1 − χ(θ2)

h(θ1, θ2) − zdθ2. (5.5)

For the same reason as above, the second integral in the right-hand side term isanalytic for |z| small for all θ1. We only need to study the integral

J (θ1, z) =∫

Td2

χ(θ2)

h(θ1, θ2) − zdθ2. (5.6)


Therefore, we use the assumptions that 0 is the unique minimum of h and thatit is quadratic nondegenerate. This implies, that for δ > 0 sufficiently small, for|θ1| < δ, the function θ2 �→ h(θ1, θ2) has a unique minimum, say θ2(θ1), thatthis minimum is quadratic nondegenerate. Let h2(θ1) be the minimal value, i.e.h2(θ1) = h(θ1, θ2(θ1)). Then, the functions θ1 �→ θ2(θ1) and θ1 �→ h(θ1) are realanalytic in |θ1| < δ.

All these statements are immediate consequences of the analytic Implicit Func-tion Theorem applied to the system of equations ∇θ2h(θ1, θ2) = 0.

So, for |θ | < δ, one can write

h(θ1, θ2) = h2(θ1) + 〈(θ2 − θ2(θ1)), Q2(θ1)(θ2 − θ2(θ1))〉 ++ O(|θ2 − θ2(θ1)|3),

where Q2(θ1) is the Hessian matrix of h(θ1, θ2) at the point θ2(θ1).We can now use the analytic Morse Lemma (see, e.g., [12]) uniformly in the

parameter θ1. That is, for some δ0 > 0 small, there exists B2(0, δ0) ⊂ U (the ballof center 0 and radius δ0 in T

d2 ) and ψ(θ1): θ2 ∈ U → ψ(θ1, θ2) ∈ B2(θ2(θ1), 2δ0),a real analytic diffeomorphism so that, for θ ∈ U ,

h(θ1, ψ(θ1, θ2)) = h2(θ1) + (θ2, Q2(θ1)θ2). (5.7)

Moreover, the Jacobian matrix of ψ at θ2(θ1) is the identity matrix, and the mappingθ1 �→ ψ(θ1) is real analytic (here, we take the norm in the Banach space of realanalytic function in a neighborhood of 0).

Before we return to the analysis of J , let us describe h2(·) and θ2(·) more pre-cisely. Let Q be the Hessian matrix of h at 0. As h has a quadratic nondegenerateminimum at 0, Q is definite positive. We can write this d × d-matrix in the form

Q =(

Q1 R∗R Q2

)

, (5.8)

where Q1,2 is the restriction of Q to Rd1,d2 when one decomposes R

d = Rd1 ×

Rd2 . Both Q1 and Q2 are positive definite; actually, the positive definiteness of Q

ensures that the matrices Q1 − R∗Q−12 R and Q2 − RQ−1

1 R∗ are positive definite.Using the Taylor expansion of h near 0, one computes

θ2(θ1) =−Q−12 Rθ1 + O(|θ1|2), Q2(θ1) = Q2 + O(|θ1|),

h2(θ1) =([Q1 − R∗Q−12 R]θ1, θ1) + O(|θ1|3).

(5.9)

Let us also note here that

DetQ = DetQ1 · Det(Q2 − R∗Q−11 R)

= DetQ2 · Det(Q1 − RQ−12 R∗). (5.10)


We now return to J . Performing the change of variables θ → ψ(θ) in J (θ1, z), weget

J (θ1, z) =∫

Td2

χ(θ1, θ2)

(θ2, Q2(θ1)θ2) + h2(θ1) − zdθ2,

where

χ(θ1, θ2) := χ(ψ(θ1, θ2)) Det(∇θ2ψ(θ1, θ2)).

(5.11)

Choosing δ sufficiently small with respect to δχ (defining χ ), we see thatχ(ψ(θ1, θ2)) = 1 for all |θ1| < δ and |θ2| < δ. Hence, the function χ(θ1, θ2)

is real analytic in a neighborhood of (0, 0).To compute the integral in the right-hand side of (5.11), we change to polar

coordinates (recall that χ is supported near 0) to obtain

J (θ1, z) = Det(Q2(θ1))−1/2

∫ +∞

0

χ(θ1, r)rd2−1

r2 + h2(θ1) − zdr, (5.12)

where

χ(θ1, r) := 1

(2π)d2

∫

Sd2−1

χ(θ1, rξ) dξ. (5.13)

The factor (2π)−d2 in the last integral comes from the fact that dθ2 denotes thenormalized Haar measure on T

d2 , i.e. the Lebesgue measure divided by (2π)d2 .Note again that (θ1, r) �→ χ(θ1, r) is real analytic in a neighborhood of 0, and

χ(θ1, 0) = 1

(2π)d2Det(∇θ2ψ(θ1, θ2(θ1))) · Vol(Sd2−1).

Moreover, as∫

Sd2 ξk dξ = 0 if k is multi-index of odd length, we known that

the Taylor expansion of χ(θ1, r) contains only even powers of r , i.e. there ex-ists a function χ(θ1, r) analytic in a neighborhood of (0, 0) such that χ(θ1, r) =χ(θ1, r

2).We now use

LEMMA 5.1. Let χ be a smooth compactly supported function such that χ be realanalytic is a neighborhood of 0. Define the integral Jχ (z) to be

Jχ (z) =∫ +∞

0

χ(r2)rn−1

r2 + zdr.

Then, one has

Jχ (z) = S(z) · H(z) + G(z), (5.14)

where

(1) G and H are real analytic in a neighborhood of 0;


(2) they satisfy H(0) = χ(0) and G(0) > 0 if χ(0) > 0 and χ � 0;(3) the function S is defined by

• if n is even, then S(z) = 12 · (−1)n/2z(n−2)/2 · log z;

• if n is odd, then S(z) = π2 · (−1)(n−1)/2z(n−1)/2 1√

z.

Here,√

z and log z denote respectively the principal determination of thesquare root and of the logarithm.

The proof of this result is elementary; after a cut-off near zero, one expands χ ina Taylor series near 0, and computes the resulting integrals term by term essentiallyexplicitly (see [15] for more details).

Putting (5.5), (5.6), (5.11), (5.12) and (5.14) together, we obtain that

I (θ1, z) = S(h2(θ1) − z) · H(θ1, h2(θ1) − z) + G(θ1, h2(θ1) − z), (5.15)

where

• S is described in point (3) of Lemma 5.1;• (θ1, z) �→ H(θ1, z) and (θ1, z) �→ G(θ1, z) are real analytic in θ1 and z in a

neighborhood of 0;• one has

H(θ1, 0) = 1

(2π)d2Det(Q2(θ1))

−1/2 · Det(∇θ2ψ(θ1, θ2)) Vol(Sd2−1)

and G(0, 0) is positive.

The last point here is obtained combining point (2) of Lemma 5.1, (5.12) and(5.13), and using the decomposition (5.5).

The first immediate consequence of (5.15) is that, if e ∈ R and h2(θ1) > e, then

I (θ1, e + iε) − I (θ1, e − iε) → 0 when ε → 0+.

This implies that, if h2(θ1) > e, one has

f (θ1, e) = 0.

Assume now that h2(θ1) � e. As 0 � h2(θ1), −e � h2(θ1) − e � 0. We now needto distinguish different cases according to the dimension d2. Consider the case

• d2 = 1: by (5.15), as H and G are analytic, one has

limε→0+ I (θ1, e + iε)

= −π

2

i√|h2(θ1) − e|H(θ1, h2(θ1) − e) + G(θ1, h2(θ1) − e).

Using again the fact that H and G are analytic and that H(θ1, 0) does notvanish for θ1 small, we get


limε→0+

1 + tI (θ1, e + iε)

1 + tI (θ1, e − iε)

= −1 + i2G(0, 0)

√|h2(θ1) − e|tH(0, 0)

+ o(√|h2(θ1) − e|

).

As G(0, 0), H(0, 0) and t are also positive, one finally obtains

f (θ1, e) = 12 [1 + O(

√|h2(θ1) − e|)] · 1{h2(θ1)�e}.

• d2 = 2: in this case, one computes

limε→0+ I (θ1, e + iε)

= 12(| log |h2(θ1) − e|| + iπ)H(θ1, h2(θ1) − e) + G(θ1, h2(θ1) − e).

Using again the fact that H and G are analytic, we get

limε→0+

1 + tI (θ1, e + iε)

1 + tI (θ1, e − iε)

=(

1 + 2iπ

| log |h2(θ1) − e||)

· (1 + O[(log |h2(θ1) − e|)−1]).

So that finally, one has

f (θ1, e) = 1

| log |h2(θ1) − e||(1 + O[(log |h2(θ1) − e|)−1]) · 1{h2(θ1)�e}.

• d2 � 3: in this case, one has to distinguish two cases namely 1 + tI (0, 0) = 0or not, as well as the case of even and odd dimensions.

Let us first assume:

– that 1 + tI (0, 0) > 0: as H and G are analytic, one has

limε→0+ I (θ1, e + iε) =

(lim

ε→0+ S(h2(θ1) − e − iε))

×

×H(θ1, h2(θ1) − e) + G(θ1, h2(θ1) − e).

As G is analytic and as S(0) = 0, one has G(θ1, h2(θ1)) = I (θ1, 0). So, forθ1 small, we know that 1 + tG(θ1, 0) �= 0. Here, we used the continuity ofG and the fact that h2(θ1) is of size |θ1|2 hence small. This gives

limε→0+

1 + tI (θ1, e + iε)

1 + tI (θ1, e − iε)= 1 + t · s(h2(θ1) − e) · H(θ1, 0)

1 + t · G(θ1, 0)· (1 + R),

where

s(x) = limε→0+[S(x − iε) − S(x + iε)],

R = O((h2(θ1) − e) · |S(h2(θ1) − e)|, (h2(θ1) − e)).

So that finally, for e small, one has


f (θ1, e) = t · s(h2(θ1) − e) · H(θ1, 0)

1 + tG(θ1, 0)1{h2(θ1)�e}(1 + R), (5.16)

where

s(x) = 12 |x|(d2−2)/2 (5.17)

and R is given above.

From these asymptotics and from (5.3), integrating f in (5.3), using (5.9) and(5.10), one gets that

• if d2 = 1:∫ E

0dNt

s (e) ∼E→0+

Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)· E1+d1/2,

(5.18)• if d2 = 2:

∫ E

0dNt

s (e) ∼E→0+

2Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)

E1+d1/2

|log E| ,

• if d2 � 3 and 1 + tI (0, 0) > 0:∫ E

0dNt

s (e) ∼E→0+

c(d1, d2) Vol(Sd2−1) Vol(Sd1−1)

d(2π)d√

DetQ· t

1 + tI (0, 0)×

×s(E)E1+d1/2, (5.19)

where Sd1,2−1 are respectively the d1,2 − 1-dimensional unit spheres, and s is given

by (5.17). Here, c(d1, d2) is the integral

c(d1, d2) =∫ 1

0rd1−1(1 − r2)(d2−2)/2 dr. (5.20)

5.1.1. The Borderline Case

Though it will not find direct applications in this paper, let us now turn to the casewhen d2 � 3 and 1+tI (0, 0) = 0. Notice that this assumption implies t < 0. When1 + tI (0, 0) = 0, one has to take a closer look at the vanishing of 1 + tG(θ1, 0)

when θ1 → 0. We will now assume that

(H): I (θ1, 0) has a local maximum for θ1 = 0.

Remark 5.1. Notice that this assumption was also necessary when we discussedfluctuating edges. Actually, in that setting, we even required that the maximum benondegenerate if d1 � 3. This seems quite natural as the case 1 + tI (0, 0) = 0 isexactly the border line between the fluctuating edges and stable edges.


Let us recall that as above, we need to compute the asymptotic when e → 0+of the integral

∫

Td1

f (θ1, e) dθ1 =∫

{h2(θ1)�e}f (θ1, e) dθ1,

where f is defined by (5.4). Using (5.9) we can find a analytic change of variableθ1 �→ ψ(θ1) such that h2(ψ

−1(θ1)) = 〈Q1θ1, θ1〉 =: q2(θ1) where Q1 = Q1 −R∗Q−1

2 R (the matrices Q1,2 and R are defined in (5.8)) and ψ(θ1) = θ1 +O(|θ1|2).So, we want to study

∫

{q2(θ1)�e}f (ψ(θ1), e)|Det∇θ1ψ(θ1)| dθ1.

Let us perform one more change of variable in the integral above, namely θ1 ↔√eθ1; hence, we need to study

∫

{q2(θ1)�1}f (ψ(

√eθ1), e)|Det∇θ1ψ(

√eθ1)| dθ1.

Notice that, for e small, on {〈Q1θ1, θ1〉 � 1}, one has

|Det∇θ1ψ(√

eθ1)| = 1 + O(√

e).

We now study f (ψ(√

eθ1), e) for e small and {q2(θ1) � 1}.Using the analyticity of G and H , for ε > 0, we start with rewriting (5.15) in

the following way

1 + tI (ψ(√

eθ1), e + iε)

= 1 + tG(ψ(√

eθ1), 0) + te∂zG(0, 0)(q2(θ1) − 1) ++ tS(e · (q2(θ1) − 1) − iε))H(ψ(

√eθ1), 0) +

+ O(ε + e2 + |e · S(e)|). (5.21)

Let us now distinguish between the different dimensions, i.e. between the casesd2 = 3, d2 = 4 and d2 � 5. Substituting the asymptotics for S given in Lemma 5.1and using the analyticity of G and H , one obtains the following:

• If d2 = 3: define F±(θ1, e) = limε→0+ 1 + tI (ψ(√

eθ1), e ± iε). Forq2(θ1) < 1, one has

F±(θ1, e) = √e(∓it (2π)−d2 |1 − q2(θ1)|1/2 Det(Q2)

−1/2 ++ t · g(θ1) + o(

√e)),

where

t · g(θ1) = lime→0+

1√e[1 + tG(ψ(

√eθ1), 0)]. (5.22)


This gives, for q2(θ1) < 1,

f (ψ(√

eθ1), e) ∼e→0+

1

πArg(−i(2π)−d2 Det(Q2)

−1/2|1 − q2(θ1)|1/2 +

+ g(θ1)).

We notice that this last argument is nonpositive. As a result we obtain that∫ E

0dNt

s (e) de ∼E→0+

∫|θ1|�1 Arg(−i|1 − θ2

1 |1/2 + g(θ1)) dθ1

d1(d1 + 2)π(2π)d1


2 R∗)· E1+d1/2,

(5.23)where

g(θ1) = (2π)d2√

Det(Q2)g((Q1 − RQ−12 R∗)−1/2θ1)

and g is defined by (5.22).

Remark 5.2. In some cases, g and g are identically vanishing. This hap-pens, for example, if h is a ‘separate variable’ function, i.e. if h(θ1, θ2) =h1(θ1) + h2(θ2). Indeed, in this case, h2(θ1) = h1(θ1) and I (θ1, h2(θ1)) =I (0, 0), hence, G does not depend on θ1, i.e. G(θ1, z) = G(z).

When g vanishes identically, formula (5.23) becomes (5.18) except for thesign which changes to −.

The integral∫|θ1|�1 Arg(−it |1−θ2

1 |1/2+g(θ1)) dθ1 is negative. Hence, com-paring (5.23) to (5.19), we see that, asymptotically when E → 0+,∫ E

0 dNts (e) de is larger when 1 + tI (0, 0) = 0 than when 1 + tI (0, 0) > 0.

This is explained by the fact that, when 1 + tI (0, 0) = 0, a zero energy reso-nance (or eigenvalue if d2 � 5) is created. This resonance (eigenvalue) carriesmore weight. Of course, the same phenomenon happens for the spectral shiftfunction.

To conclude the case d2 = 3, let us notice that we did not use assump-tion (H).

• If d2 = 4: let us start with computing ∂θ1G(0, 0). Therefore, we use G(θ1, 0) =I (θ1, h2(θ1)) and compute

∂θ1G(0, 0) = ∂θ1[I (θ1, h2(θ1))]|θ1=0

= −(∫

Td2

∂θ1(h(θ1, θ2) − h2(θ1))

(h(θ1, θ2) − h2(θ1))2dθ2

)

|θ1=0

= −∫

Td2

∂θ1h(0, θ2)

(h(0, θ2))2dθ2 = 0

as 0 is a local maximum of I (θ1, 0). This computation immediately gives that1 + tG(ψ(

√eθ1), 0) = O(e). Hence, Equation (5.21) gives


F±(θ1, e) = te(q2(θ1) − 1)(log e + log |q2(θ1) − 1| + h(e)) ×

×(

1 + ∓iπ

log e + log |q2(θ1) − 1| + h(e)

)

,

where h(e) is bounded and does not depend on the sign ±. This gives, forq2(θ1) < 1,

f (ψ(√

eθ1), e) ∼e→0+ − 1

|log e| .Integrating over θ1 and e, we obtain

∫ E

0dNt

s (e) ∼E→0+ − 2Vol(Sd1−1)

d1(d1 + 2)(2π)d1


2 R∗)

E1+d1/2

|log E| .

• If 1 + tI (0, 0) = 0 and d2 � 5: we now compute ∂2θ1

Q(0, 0). Therefore, wecontinue the computation done above to obtain

∂2θ1

G(0, 0) = −∂θ1

(∫

Td2

∂θ1(h(θ1, θ2) − h2(θ1))

(h(θ1, θ2) − h2(θ1))2dθ2

)

|θ1=0

= −(∫

Td2

∂2θ1

(h(θ1, θ2) − h2(θ1))

(h(θ1, θ2) − h2(θ1))2dθ2

)

|θ1=0

+

+ 2

(∫

Td2

[∂θ1(h(θ1, θ2) − h2(θ1))]2

(h(θ1, θ2) − h2(θ1))3dθ2

)

|θ1=0

= −∫

Td2

∂2θ1

h(0, θ2)

(h(0, θ2))2dθ2 + 2

∫

Td2

[∂θ1h(0, θ2)]2

(h(0, θ2))3dθ2 +

+(∫

Td2

1

(h(0, θ2))2dθ2

)

Q2, (5.24)

where Q2 is defined in (5.8). On the other hand, one has

∂zG(0, 0) = −∂zI (0, z)|z=0 = −J where J :=∫

Td2

1

(h(0, θ2))2dθ1.

Plugging this and (5.24) into (5.21), we obtain

1 + tI (ψ(√

eθ1), e ± iε)

= −teJ + o(e) + tS(e · (q2(θ1) − 1) ∓ iε)(H(0, 0) + o(1)),

where o(e) does not depend of ±. This gives, for q2(θ1) < 1,

f (ψ(√

eθ1), e) ∼e→0+ −s(e · (q2(θ1) − 1))

J.

Integrating over θ1 and e, we obtain


∫ E

0dNt

s (e) ∼E→0+

c(d1, d2) Vol(Sd2−1) Vol(Sd1−1)

d(2π)d√

DetQ· −1

J· s(E)Ed1/2,

where c(d1, d2) is defined in (5.20).

Appendix

Pick E < −d. We now prove that, for h taken as in Remark 1.1, the function h

defined in (1.7) is not constant. For the purpose of this argument, we write θ1 =(θ1, . . . , θd1). To check that h is not constant, by (1.2) and (1.7), it suffices tocheck that the function θ1 �→ I (θ1, E) is not constant, hence, that the functionθ1 �→ J (θ1) defined by

J (θ1) = 1

(2π)d1−1

∫

[0,2π]d1−1I (θ1, θ2, . . . , θd1, E) dθ2 . . . dθd1

= 1

(2π)d−1

∫

[0,2π]d−1

1

h(θ1, θ ′) − Edθ ′ (A.1)

is not constant. We used the notation θ = (θ1, θ2) = (θ1, θ ′).Recall from Remark 1.1 that h(θ) = h0(G

′ · θ) where G′ ∈ GSL(Z) and h0 isdefined in (1.1). So, the nth Fourier coefficient of J is given by

Jn = 1

(2π)d

∫

[0,2π]

∫

[0,2π]d−1

einθ1

h(θ1, θ ′) − Edθ ′ dθ1

= 1

(2π)d

∫

[0,2π]deinθ1

h0(G′ · θ) − Edθ

= 1

(2π)d

∫

[0,2π]dein(G′−1·θ)1

h0(θ) − Edθ = ein(G′−1·θπ )1

(2π)d

∫

[−π,π]dein(G′−1·θ)1

−h0(θ) − Edθ,

where (G′−1·θ)1 denotes the first coordinate of the vector G′−1·θ , and θπ , the vector(π, . . . , π) in R

d . So to prove that Jn does not vanish for any n which implies thatJ is not constant, it suffices to prove that the Fourier coefficients of (h0(θ) − E)−1

do not vanish. This is a consequence of the Neuman expansion

1

−h0(θ) − E= −1

E

∑

k�0

(h0(θ)

−E

)k

.

Indeed, the nth Fourier coefficient in each of the terms of order k larger than n inthis series is positive: it is easily seen as −E > 0 and the multiplication opera-tor (h0)

n is unitarily equivalent through Fourier transformation to (− 12�)n; so the

Fourier coefficients of (h0)n are the entries of the zeroth row of the matrix (− 1

2�)n

and, the n first super- and subdiagonals of this convolution matrix are positive.


References

1. Birman, M. Sh. and Yafaev, D. R.: The spectral shift function. The papers of M. G. Kreın andtheir further development, Algebra i Analiz 4(5) (1992), 1–44.

2. Chahrour, A.: Densité intégrée d’états surfaciques et fonction généralisée de déplacement spec-tral pour un opérateur de Schrödinger surfacique ergodique, Helv. Phys. Acta 72(2) (1999),93–122.

3. Chahrour, A. and Sahbani, J.: On the spectral and scattering theory of the Schrödinger operatorwith surface potential, Rev. Math. Phys. 12(4) (2000), 561–573.

4. Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.: Schrödinger Operators, Springer-Verlag, Berlin, 1987.

5. Dembo, A. and Zeitouni, O.: Large Deviation Techniques and Applications, Jones and BartlettPublishers, Boston, 1992.

6. Dimassi, M. and Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit, In: LondonMath. Soc. Lecture Note Ser. 268, Cambridge University Press, Cambridge, 1999.

7. Durrett, R.: Probability: Theory and Examples, 2nd edn, Duxbury Press, Belmont, CA, 1996.8. Englisch, H., Kirsch, W., Schröder, M. and Simon, B.: Density of surface states in discrete

models, Phys. Rev. Lett. 61(11) (1988), 1261–1262.9. Englisch, H., Kirsch, W., Schröder, M. and Simon, B.: Random Hamiltonians ergodic in all but

one direction, Comm. Math. Phys. 128(3) (1990), 613–625.10. Helffer, B. and Sjöstrand, J.: On diamagnetism and the De Haas–Van Alphen effect, Ann. Inst.

H. Poincaré Phys. Théor. 52 (1990), 303–375.11. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Springer-Verlag, Heidel-

berg, 1983.12. Hörmander, L.: The Analysis of Linear Partial Differential Equations. I, Grundlehren Math.

Wiss. 256, Springer-Verlag, 1990.13. Jakšic, V. and Last, Y.: Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12(11) (2000),

1465–1503.14. Kirsch, W.: Random Schrödinger operators, In: A. Jensen and H. Holden (eds), Schrödinger

Operators, Lecture Notes in Phys. 345, Springer-Verlag, Berlin, 1989. Proceedings, Sonder-borg, Denmark, 1988.

15. Klopp, F.: Resonances for perturbations of a semi-classical periodic Schrödinger operator, Ark.Mat. 32 (1994), 323–371.

16. Klopp, F.: Band edge behaviour for the integrated density of states of random Jacobi matricesin dimension 1, J. Statist. Phys. 90(3–4) (1998), 927–947.

17. Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators,Duke Math. J. 98(2) (1999), 335–396.

18. Klopp, F.: Weak disorder localization and Lifshitz tails, Comm. Math. Phys. 232 (2002), 125–155.

19. Klopp, F. and Wolff, T.: Lifshitz tails for 2-dimensional random Schrödinger operators, J. Anal.Math. 88 (2002), 63–147.

20. Kostrykin, V. and Schrader, R.: The density of states and the spectral shift density of randomSchrödinger operators, Rev. Math. Phys. 12(6) (2000), 807–847.

21. Kostrykin, V. and Schrader, R.: Regularity of the surface density of states, J. Funct. Anal. 187(1)(2001), 227–246.

22. Mather, J. N.: On Nirenberg’s proof of Malgrange’s preparation theorem, In: Proceedings ofLiverpool Singularities-Symposium I, Lecture Notes in Math. 192, Springer-Verlag, Berlin,1971.

23. Pastur, L. and Figotin, A.: Spectra of Random and almost-Periodic Operators, GrundlehrenMath. Wiss. 297, Springer-Verlag, Berlin, 1992.


24. Simon, B.: Spectral analysis of rank one perturbations and applications, In: Mathemati-cal Quantum Theory. II. Schrödinger Operators (Vancouver, BC, 1993), Amer. Math. Soc.,Providence, RI, 1995, pp. 109–149.

25. Yafaev, D. R.: Mathematical Scattering Theory, Trans. Math. Monogr. 105, Amer. Math. Soc.,Providence, RI, 1992.

Degenerate Hermitian Manifolds

FAZILET ERKEKO~GLU

Department of Mathematics, Hacettepe University, Beytepe, Ankara 06532, Turkey.e-mail: [email protected]

(Received: 9 June 2004; in final form: 1 September 2005)

Abstract. The geometry of almost complex manifolds with degenerate indefinite Hermitian

metrics is studied.

Mathematics Subject Classifications (2000): 53C15, 53C50, 53C55.

Key words: Ka€hler manifold, almost complex manifold, almost symplectic manifold, singular

semi-Riemannian manifold, Koszul connection, Hermitian connection, Nijenhuis torsion tensor.

1. Introduction

Let g be a Riemannian metric, � be an almost symplectic structure and J be an

almost complex structure on a manifold M. The differential geometric structures

g, � and J are called compatible if g(X, Y) = �(X, JY) for every X, Y 2 G (TM). A

manifold with compatible differential geometric structures g, � and J is called a

Ka€hler manifold provided that � is closed and J is the canonical almost complex

structure induced by some complex manifold structure on M.

The purpose of this paper is to study the Ka€hler geometry of almost complex

manifolds with degenerate indefinite Hermitian metrics in their complex tangent

bundles. That is, we shall allow the differential geometric structures g and �to be a degenerate metric tensor and a degenerate almost symplectic structure

respectively.

The geometry of manifolds with degenerate indefinite metrics has been

studied in [2]. It is shown that a manifold M with a degenerate indefinite metric gadmits a geometric structure iff g is Lie parallel along the degenerate vector fields

on M. In this case, we call (M, g) a singular semi-Riemannian manifold. Then it

is possible to attach a nondegenerate tangent bundle to (M, g) which admits a

connection whose curvature tensor satisfies the usual identities of the curvature

tensor of a Levi-Civita connection. We call this connection the Koszul connection

of (M, g). In our study of degenerate Ka€hler manifolds we shall replace the

Levi-Civita connection with the Koszul connection in the associated nondegen-

erate tangent bundle and replace the curvature tensor with the intrinsic curvature

tensor of the associated nondegenerate tangent bundle.

Mathematical Physics, Analysis and Geometry (2005) 8: 361Y387

DOI: 10.1007/s11040-005-9001-9

# Springer 2006

The plan of this paper is as follows: In Section 2, we shall give the main

definitions and investigate the properties of compatible degenerate differential

geometric structures on a manifold. In Section 3, we shall study the induced

nondegenerate differential geometric structures in the associated nondegenerate

tangent bundle. In Section 4, we shall obtain the fundamental relations among the

nondegenerate differential geometric structures and define the degenerate Ka€hler

manifolds. (This definition does not necessarily require a degenerate Ka€hler

manifold to be a complex manifold to possess Ka€hler geometry.) In Section 5,

we shall study the curvature of degenerate Ka€hler manifolds. In Section 6, we

shall investigate the relation of the Koszul connection of a complex degenerate

Ka€hler manifold to the unique Hermitian connection of type (1,0) in its asso-

ciated nondegenerate holomorphic tangent bundle.

This paper will also improve some of the results in [2]. In particular, in Section

4 we shall prove the existence and uniqueness of the Koszul connection of a

manifold (M, g) and, in Section 5, we obtain the second Bianchi identity for the

intrinsic curvature tensor of (M, g) and prove the Schur lemma for degenerate

manifolds.

2. Degenerate Differential Geometric Structures

Here we briefly state the main concepts and definitions used throughout this

paper.

DEFINITION 2.1. Let V be a real vector space.

(a) A symmetric bilinear form g on V is called an inner product on V. The type

(k, m, n) of an inner product g on V is defined by

k ¼ dimfu 2 V j gðu; vÞ ¼ 0; 8v 2 Vg (the nullity of g),

m ¼ supfdimW jW � V 3 gðw;wÞ < 0; 80 6¼ w 2 Wg (The index of g), and

n ¼ supfdimW jW � V 3 gðw;wÞ > 0;80 6¼ w 2 Wg:

An inner product g on V is called nondegenerate if k = 0. (V, g) is called an inner

product space of type (k, m, n) if g is an inner product on V of type (k, m, n).

(b) A skew-symmetric bilinear form � on V is called a symplectic form on V.

The nullity k of a symplectic form � on V is defined by

k ¼ dimfu 2 Vj�ðu; vÞ ¼ 0; 8v 2 Vg:A symplectic form � on V is called nondegenerate if k = 0. (V, �) is

called a symplectic vector space with nullity k if � is a symplectic form on

V with nullity k.

362 FAZILET ERKEKO~GLU

DEFINITION 2.2. Let M be a smooth manifold.

(a) A smooth g 2 G (TM* � TM*) is called a metric tensor of type (k, m, n) on

M if, 8p 2 M, (TpM, gp) is an inner product space of constant type (k, m, n).

(M, g) is called a manifold with metric tensor g of type (k, m, n) if g is a

metric tensor on M of type (k, m, n).

(b) A smooth � 2 G (TM* � TM*) is called an almost symplectic structure on

M with nullity k if, 8p 2 M, (TpM, �p) is a symplectic vector space with

constant nullity k. (M, �) is called an almost symplectic manifold with

nullity k if � is an almost symplectic structure on M with nullity k. An

almost symplectic manifold (M, �) with nullity k is called a symplectic

manifold of nullity k if d� = 0.

EXAMPLE 2.3. Let f: S3 Y S2 be the Hopf fibration of S3 over S2. Let g1 be the

standard Riemannian metric on S2 and �1 be the Riemannian volume form of S2.

Then (S3, g = f * g1) is a manifold with metric g of type (1, 0, 2) and (S3, � =

f * �1) is an almost symplectic manifold with nullity 1. In fact, it is easy to check

that (S3, �) is a symplectic manifold with nullity 1.

EXAMPLE 2.4. Let M = N � H, where N is a manifold and H is a semi-

Riemannian manifold with metric h of type (0, m, n) which admits a

nondegenerate almost symplectic structure �H. Let P: M Y N and Q: M Y Hbe the projections onto N and H respectively. Then (M, g = Q* h) is a manifold

with metric g of type (k, m, n), where k = dimN, and (M, � = Q* �H) is an almost

symplectic manifold with nullity k = dimN. In fact, it is easy to check that (M, �)

is a symplectic manifold with nullity k = dimN.

DEFINITION 2.5. Let g be a metric tensor of type (k, m, n) and let � be an

almost symplectic structure with nullity k on a manifold M.

(a) The degenerate bundle M?g of (M, g) is defined by

M?g ¼[

p2M

fu 2 TpM j gðu; vÞ ¼ 0;8v 2 TpMg:

(M, g) is called integrable if M?g is an integrable subbundle of TM. (M, g) is

called a singular semi-Riemannian manifold if LUg ¼ 0 for every U 2�ðM?g Þ where L is the Lie derivative on M.

(b) The degenerate bundle M?� of (M, �) is defined by

M?� ¼[

p2M

fu 2 TpM j �ðu; vÞ ¼ 0; 8v 2 TpMg:

(M, �) is called integrable if M? � is an integrable subbundle of TM. (M,

�) is called a singular almost symplectic manifold if LU� ¼ 0 for every

U 2 �ðM?�Þ.

DEGENERATE HERMITIAN MANIFOLDS 363

PROPOSITION 2.6. If (M, g) is a singular semi-Riemannian manifold (resp.,(M, �) is a singular almost symplectic manifold) then (M, g) (resp., (M, �)) isintegrable.

Proof. To show (M, g) is integrable, let U1;U2 2 �ðM?g Þ. Then, since LU1g ¼

0 for every X 2 G(TM),

gð½U1;U2�;XÞ ¼ U1gðU2;XÞ � gðU2; ½U1;X�Þ � ðLU1gÞðU2;XÞ

¼ 0:

Hence, ½U1;U2� 2 �ðM?g Þ. Integrability of (M, �) can be similarly shown. Ì

PROPOSITION 2.7. Let (M, �) be an almost symplectic manifold. Then (M, �)

is a singular almost symplectic manifold iff {(U)d� = 0 for every U 2 �ðM?�Þ,where { is the interior product. In particular, every symplectic manifold is asingular symplectic manifold.

Proof. Let U 2 �ðM?�Þ;X;Y 2 �ðTMÞ. Then,

ðLU�ÞðX; YÞ ¼ ð�ðUÞd�ÞðX;YÞ þ ðd�ðUÞ�ÞðX; YÞ¼ ð�ðUÞd�ÞðX;YÞ:

Hence, LU� ¼ 0 iff {ðUÞd� ¼ 0. Ì

We recall that J 2 G(Hom(TM, TM)) on a manifold M is called an almost

complex structure on M if J2 = jid. A manifold (M, J) with an almost complex

structure J is called an almost complex manifold. An almost complex structure

on a manifold M is called integrable if it is the canonical almost complex

structure induced by some complex manifold structure on M. The Nijenhuis

torsion tensor N of an almost complex structure J on a manifold M is defined by

NðX; YÞ ¼ ½X; Y� þ J½JX;Y� þ J½X; JY� � ½JX; JY�;

where X, Y 2 G(TM). It is well known that an almost complex structure on a

manifold M is integrable iff N = 0 (cf. p. 261 of [4]).

DEFINITION 2.8. Let g be a metric tensor of type (k, m, n), � be an almost

symplectic structure with nullity k0 and J be an almost complex structure on a

manifold M. The differential geometric structures g, � and J on M are called

compatible if g(X, Y) = �(X, JY) for every X, Y 2 G(TM).

THEOREM 2.9. Let g, � and J be compatible differential geometric structureson a manifold M. Then M?g ¼ M?�ð¼ M?Þ and M? is invariant under J. Inparticular,

gðJX; JYÞ ¼ gðX;YÞ; Y�ðJX; JYÞ ¼ �ðX;YÞ and �ðX; YÞ ¼ gðJX; YÞ

for every X, Y 2 G(TM).


Proof. Since J is nonsingular, X 2 �ðM?g Þ iff X 2 �ðM?�Þ. Hence, M?g ¼M?�ð¼ M?Þ. Also, for each Y 2 �ðM?Þ, since gðX;YÞ ¼ �ðX; JYÞ ¼ 0 for every

X 2 �ðTMÞ; JY 2 �ðM?Þ. Thus, M? is invariant under J. The rest of the claim is

now immediate. Ì

From now on, if g, � and J are compatible differential geometric structures on

a manifold M then we shall denote M?g ¼ M?� by M?. Note also that if either M?gor M?� is integrable then M? is integrable.

COROLLARY 2.10. Let g, � and J be compatible differential geometricstructures on a manifold M. Then,

(a) If LUg ¼ 0 and LU� ¼ 0 for every U 2 �ðM?Þ then LUJ 2 �ðHomðTM;M?ÞÞ for every U 2 �ðM?Þ.

(b) If LUg ¼ 0 and LUJ 2 �ðHomðTM;M?ÞÞ for every U 2 �ðM?Þ thenLU� ¼ 0 for every U 2 �ðM?Þ.

(c) If LU� ¼ 0 and LUJ 2 �ðHomðTM;M?ÞÞ for every U 2 �ðM?Þ thenLUg ¼ 0 for every U 2 �ðM?Þ.

Proof. For every U 2 �ðM?Þ;X; Y 2 �ðTMÞ,ðLUgÞðX; YÞ ¼ UgðX; YÞ � gðLUX;YÞ � gðX;LUYÞ

¼ U�ðX; JYÞ � �ðLUX; JYÞ � �ðX; JðLUYÞÞ¼ ðLU�ÞðX; JYÞ þ �ðX;LUðJYÞÞ � �ðX; JðLUYÞÞ¼ ðLU�ÞðX; JYÞ þ �ðX; ðLUJÞYÞ:

Thus, since J is nonsingular, (a), (b) and (c) follows from the above equation. Ì

PROPOSITION 2.11. Let g be a metric tensor and J be an almost complexstructure on a manifold M. If g(JX, JY) = g(X, Y) for every X, Y 2 G(TM) then�(X, Y) = g(JX, Y) is the unique symplectic structure � on M such that g, � and Jare compatible differential geometric structures.

Proof. Immediate from Theorem 2.9 once we show that � is skew-

symmetric. Indeed, for every X, Y 2 G(TM),

�ðX;YÞ ¼ gðJX; YÞ¼ gðJ2X; JYÞ¼ �gðX; JYÞ¼ �gðJY;XÞ¼ ��ðY;XÞ: Ì

PROPOSITION 2.12. Let � be an almost symplectic structure and J be analmost complex structure on a manifold M. If �(JX, JY) = �(X, Y) for every X,

Y 2 G(TM) then g(X, Y) = �(X, JY) is the unique metric tensor on M such that g,

� and J are compatible.

Proof. It can be proven similar to Proposition 2.11. Ì


3. Associate Nondegenerate Differential Geometric Structures

In this section and thereafter, let g be a metric tensor, � be an almost symplectic

structure and J be an almost complex structure on a manifold M.

DEFINITION 3.1. Let g, � and J be compatible differential geometric

structures on a manifold M. The associated nondegenerate tangent bundle TMwith differential geometric structures �gg; �� and �JJ are defined as follows: TM ¼TM=M? and �: TM! TM is the canonical projection. �ggð �XX; �YYÞ ¼ gðX;YÞ, where

X, Y 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY. ��ð �XX; �YYÞ ¼ �ðX; YÞ, where X;Y 2�ðTMÞ with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY. �JJ �XX ¼ �ðJXÞ, where X 2 G(TM) with

�ðXÞ ¼ �XX.

It is easy to check that TM; �gg; �� and �JJ are well defined (cf. Theorem 2.9).

Note that �gg is a nondegenerate metric, �� is a nondegenerate symplectic form and�JJ is an almost complex structure in TM. Furthermore, it follows from Theorem

2.9 that �gg; �� and �JJ are compatible, that is, �ggð �XX; �YYÞ ¼ ��ð �XX; �JJ �YYÞ, where �XX; �YY 2�ðTMÞ:

Remark 3.2. rankðTMÞ and indexð�ggÞ are even numbers.

Remark 3.3. If either g or � is nonsingular then TM is canonically isomorphic

to TM.

Remark 3.4. TM has an induced complex vector bundle structure from �JJ:Namely, the multiplication with complex functions on M are defined by

ðf þ ihÞ �XX ¼ f �XX þ h�JJ �XX, where �XX 2 �ðTMÞ; i ¼ffiffiffiffiffiffiffi�1p

and f, h are real valued

functions on M. Also, the canonical projection P is a complex bundle homomor-

phism between complex vector bundles (TM, J) and ðTM; �JJÞ since �JJ �� ¼ � � J.

Remark 3.5. The local Hermitian basis in TM is defined as follows: Let

rankðTMÞ ¼ 2r and indexð�ggÞ ¼ 2s. Then the basis

B ¼ f �XX1; � � � ; �XXs; �JJ �XX1; � � � ; �JJ �XXs; �XX2sþ1; � � � ; �XXsþr; �JJ �XX2sþ1; � � � ; �JJ �XXsþrg

is called a local Hermitian basis in TM, where f �XXig is a local semi-Riemannian

basis in ðTM; �ggÞ such that �JJ �XXi ¼ �XXiþs for 1 r i r s and �JJ �XXi ¼ �XXiþr�s for 2s + 1 ri r s + r. Also, if B* ¼ f�!!ig is a metrically equivalent basis to B in TM*, then

the nondegenerate symplectic structure �� in TM with respect to the basis B* is

given by

�� ¼ �Xs

i¼1

�!!i ^ �!!iþs þXsþr

i¼2sþ1

�!!i ^ �!!iþr�s:


DEFINITION 3.6. Let g, � and J be compatible differential geometric struc-

tures on a manifold M with integrable M?. The Lie derivative �LLu�XX of �XX 2 �ðTMÞ

in the direction of u 2 M? is defined by �LLu�XX ¼ �ðLUXÞjp; where X 2 G(TM)

with �ðXÞ ¼ �XX and U 2 �ðM?Þ with Up = u.

The well-definedness of �LLu can similarly be shown as in Definition 4.2 of [2].

Remark 3.7. Notice also in the above definition that �LLu is tensorial in

u 2 M?.

DEFINITION 3.8. Let g, � and J be the compatible differential geometric

structures on a manifold M with integrable M? and let �gg; �� and �JJ be the induced

differential geometric structures in TM. Then,

(a) The Lie derivative �LLu�gg of �gg in the direction of u 2 M?p is defined by

ð�LLu�ggÞð�xx; �yyÞ ¼ ðLUgÞðx; yÞ; where x, y 2 TpM with �ðxÞ ¼ �xx;�ðyÞ ¼ �yy and

U 2 �ðM?Þ with Up = u.

(b) The Lie derivative �LLu�� of �� in the direction of u 2 M?p is defined by

ð�LLu��Þð�xx; �yyÞ ¼ ðLU�Þðx; yÞ, where x; y 2 TpM with �ðxÞ ¼ �xx;�ðyÞ ¼ �yy and

U 2 �ðM?Þ with Up = u.

(c) The Lie derivative �LLu�JJ of �JJ in the direction of u 2 M?p is defined by

ð�LLu�JJÞ�xx ¼ �ððLUJÞxÞ, where x 2 TpM with �ðxÞ ¼ �xx and U 2 �ðM?Þ with

Up = u.

It can be shown as in Definition 4.3 of [2] that �LLu�gg and �LLu�� are well-defined.

We shall only show that �LLu�JJ is well-defined. To show this, let X1, X2 2 G(TM) be

such that X2 = X1 + U with �ðX2Þjp ¼ �xx ¼ �ðX1Þjp; where U 2 �ðM?Þ; and

U1;U2 2 �ðM?Þ be extensions of u. Thus U2 = U1 + U0, where U0 2 �ðM?Þ:Then,

�ððLU2JÞX2Þjp ¼ �ððLU1þU0JÞðX1 þ UÞÞjp

¼ �ððLU1JÞX1 þ ðLU1

JÞU þ ðLU0JÞX1 þ ðLU0JÞUÞjp:

But since M? is integrable and invariant under J,

�ððLU1JÞUÞ ¼ �ðLU1

ðJUÞ � JðLU1UÞÞ ¼ 0;

�ððLU0JÞUÞ ¼ �ðLU0 ðJUÞ � JðLU0UÞÞ ¼ 0;

and

�ððLU0JÞX1Þjp ¼ �ðLU0 ðJX1Þ � JðLU0X1ÞÞjp¼ �LL0ð�JJ �XX1Þ � �JJð�LL0

�XX1Þ¼ 0;

where �ðX1Þ ¼ �XX1: Thus, �ððLU2JÞX2Þjp ¼ �ððLU1

JÞX1Þjp.


THEOREM 3.9. Let g, � and J be compatible differential geometric structureson a manifold M with integrable M?. Then, for every u 2 M? and �XX; �YY 2 �ðTMÞ;(a) ð�LLu�ggÞð �XX; �YYÞ ¼ u�ggð �XX; �YYÞ � �ggð�LLu

�XX; �YYÞ � �ggð �XX; �LLu�YYÞ;

(b) ð�LLu��Þð �XX; �YYÞ ¼ u��ð �XX; �YYÞ � ��ð�LLu

�XX; �YYÞ � ��ð �XX; �LLu�YYÞ;

(c) ð�LLu�JJÞ �XX ¼ �LLuð�JJ �XXÞ � �JJð�LLu

�XXÞ:Proof. (a) and (b) can be similarly shown as in Theorem 4.4 of [2]. To prove

(c), let X 2 G(TM) with �ðXÞ ¼ �XX and U 2 �ðM?Þ with Up = u, where u 2 M?p :Then,

ð�LLu�JJÞ �XX ¼ �ððLUJÞXÞjp¼ �ðLUðJXÞ � JðLUXÞÞjp¼ �LLuð�JJ �XXÞ � �JJð�LLu

�XXÞ Ì

COROLLARY 3.10. Let g, � and J be compatible differential geometricstructures on a manifold M with integrable M?: Then,

(a) (M, g) is a singular semi-Riemannian manifold iff �LLu�gg ¼ 0 for every u 2 M?:(b) (M, �) is a singular almost symplectic manifold iff �LLu

�� ¼ 0 for everyu 2 M?:

(c) If �LLu�gg ¼ 0 and �LLu�� ¼ 0 for every u 2 M? then �LLu

�JJ ¼ 0 for every u 2 M?:(d) If �LLu�gg ¼ 0 and �LLu

�JJ ¼ 0 for every u 2 M? then �LLu�� ¼ 0 for every u 2 M?:

(e) If �LLu�� ¼ 0 and �LLu

�JJ ¼ 0 for every u 2 M? then �LLu�gg ¼ 0 for every u 2 M?:

Proof. Immediate from Definition 2.5 and Corollary 2.10. Ì

DEFINITION 3.11. Let g, � and J be compatible differential geometric struc-

tures on a manifold M, where (M, g) is a singular semi-Riemannian and (M, �) is

a singular almost symplectic manifold.

(a) The exterior derivative �dd�� of �� is defined by �dd��ð �XX; �YY; �ZZÞ ¼ d�ðX; Y; ZÞ;where X, Y, Z 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ.

(b) The Nijenhuis torsion tensor �NN of �JJ is defined by �NNð �XX; �YYÞ ¼ �ðNðX;YÞÞ:where X, Y 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY and N is the Nijenhuis

torsion tensor of J.

To show that �dd�� is well-defined, let X1, X2, Y1, Y2, Z1, Z2 2 G(TM) with

�ðX1Þ ¼ �XX ¼ �ðX2Þ;�ðY1Þ ¼ �YY ¼ �ðY2Þ;�ðZ1Þ ¼ �ZZ ¼ �ðZ2Þ: Then, X2 = X1 +

U1, Y2 = Y1 + U2, Z2 = Z1 + U3, where U1;U2;U3 2 �ðM?Þ: Hence

d�ðX2;Y2; Z2Þ ¼ d�ðX1 þ U1;Y1 þ U2;Z1 þ U3Þ¼ d�ðX1;Y1; Z1Þ;

since 1ðUÞd� ¼ 0 for every U 2 �ðM?Þ from Proposition 2.7 where 1 is the

interior product.


To show that �NN is well-defined, let X1, X2, Y1, Y2 2 G(TM) with �ðX1Þ ¼�XX ¼ �ðX2Þ;�ðY1Þ ¼ �YY ¼ �ðY2Þ. Thus, X2 = X1 + U1 and Y2 = Y1 + U2, where

U1;U2 2 �ðM?Þ: Then, since M? is integrable and LUJ 2 �ðHom ðTM;M?ÞÞ for

every U 2 �ðM?Þ, (cf. Corollary 2.10),

P([X2, Y2] + J[JX2, Y2] + J[X2, JY2] j [JX2, JY2]

= P([X1 + U1, Y1 + U2] + J[J(X1 + U1), Y1 + U2] +

+ J[X1 + U1, J(Y1 + U2)] j [J(X1 + U1), J(Y1 + U2)])

= P([X1, Y1] + [X1, U2] + [U1, Y1] + [U1, U2] +

+ J[JX1, Y1] + J[JX1, U2] + J[JU1, Y1] + J[JU1, U2] + J[X1, JY1] +

+ J[X1,JU2] + J[U1, JY1] + J[U1, JU2] j [JX1, JY1] j

j [JX1, JU2] j [JU1, JY1] j [JU1, JU2])

= P[X1, Y1] + J[JX1, Y1] + J[X1, JY1] j [JX1, JY1])

since

P([U1, U2]) = P(J[JU1, U2]) = P(J[U1, JU2]) = P([JU1, JU2]) = 0,

�ð½X1;U2� þ J½JX1;U2�Þ ¼ ��ðJððLU2JÞX1ÞÞ ¼ 0;

�ð½U1;Y1� þ J½U1; JY1�Þ ¼ �ðJððLU1JÞY1ÞÞ ¼ 0;

�ðJ½JU1;Y1� � ½JU1; JY1�Þ ¼ ��ððLJU1JÞY1Þ ¼ 0;

�ðJ½X1; JU2� � ½JX1; JU2�Þ ¼ �ððLJU2JÞX1Þ ¼ 0:

4. Degenerate Ka€hler Manifolds

We first recall some fundamental facts about singular semi-Riemannian

manifolds from [2]. Let (M, g) be a manifold with metric g of type (k, m, n).

A function l: G(TM) � G(TM) Y G(TM) is called a Koszul derivative on (M, g)

if, for every X, Y 2 G(TM), lXY is a smooth section of TM and for every X, Y, Z,

U, W 2 G(TM), f 2 CV(M),

(a) g(lU+ W X, Z) = g(lU X, Z) + g(lW X, Z)

(b) g(lfU X, Z) = fg(lU X, Z)

(c) g(lU (X + Y), Z) = g(lU X, Z) + g(lU Y, Z)

(d) g(lU fX, Z) = U(f) g(X, Z) + fg(lU X, Z)

(e) Zg(X, Y) = g(lZ X, Y) + g(X, lZ, Y)

(f) g(lX Y, Z) j g(lY X, Z) = g([X, Y], Z)

It can be shown that a manifold (M, g) admits a Koszul derivative iff (M, g) is

a singular semi-Riemannian manifold (cf. Theorem 3.4 of [2]). The nondegen-

erate tangent bundle ðTMg; �ggÞ over a manifold (M, g) is defined by TMg ¼TM=M?g and �ggð�xx; �yyÞ ¼ gðx; yÞ; where x, y 2 TM with �gðxÞ ¼ �xx;�gðyÞ ¼ �yy and

�g: TM! TMg is the canonical projection. If (M, g) is a singular semi-

Riemannian manifold, then the Koszul connection �r in ðTMg; �ggÞ is defined by�rX

�YY ¼ �gðrXYÞ, where Y 2 G(TM) with �gðYÞ ¼ �YY and l is a Koszul deriv-


ative on M (cf. Definition 4.6 of [2]). In fact we have the following existence and

uniqueness theorem.

THEOREM 4.1. Let (M, g) be a manifold with metric tensor g. Then there exista unique semi-Riemannian connection �r in ðTMg; �ggÞ satisfying �rX

�YY � �rY�XX ¼

�gð½X; Y�Þ for every X, Y 2 G(TM) with �gðXÞ ¼ �XX;�gðYÞ ¼ �YY iff (M, g) is asingular semi-Riemannian manifold. In fact, �r is the Koszul connection inðTMg; �ggÞ;

Proof. Assume that (M, g) is a singular semi-Riemannian manifold. Then, by

definition, the Koszul connection �r in ðTMg; �ggÞ is such a connection. To show its

uniqueness, let X, Y, Z 2 G(TM) with �gðXÞ ¼ �XX;�gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ:Then,

�ggð �rX�YY; �ZZÞ ¼ X�ggð �YY; �ZZÞ � �ggð �YY; �rX

�ZZÞ

¼ X�ggð �YY; �ZZÞ � �ggð �YY; �rZ�XXÞ � �ggð �YY;�gð½X;Z�ÞÞ

¼ X�ggð �YY; �ZZÞ � Z�ggð �YY; �XXÞ þ �ggð �rZ�YY; �XXÞ � �ggð �YY;�gð½X; Z�ÞÞ

¼ X�ggð �YY; �ZZÞ � Z�ggð �YY; �XXÞ þ �ggð �rY�ZZ; �XX þ �ggð�gð½Z;Y�; �XXÞ �

� �ggð �YY;�gð½X; Z�ÞÞ

¼ X�ggð �YY; �ZZÞ � Z�ggð �YY; �XXÞ þ Y�ggð �ZZ; �XXÞ � �ggð �ZZ; �rY�XXÞþ

þ �ggð�gð½Z; Y�Þ; �XXÞ � �ggð �YY;�gð½X;Z�ÞÞ

¼ X�ggð �YY; �ZZÞ � Z�ggð �YY; �XXÞ þ Y�ggð �ZZ; �XXÞ � �ggð �ZZ; �rX�YYÞ�

� �ggð�gð½Y;X�Þ; �ZZÞ þ �ggð�gð½Z; Y�; �XXÞ � �ggð �YY;�gð½X;Z�ÞÞ:

Thus,

�ggð �rX�YY; �ZZÞ ¼ 1=2fX�ggð �YY; �ZZÞ � Z�ggð �YY; �XXÞ þ Y�ggð �ZZ; �XXÞ � �ggð�gð½Y;X�Þ; �ZZÞþ

þ �ggð�gð½Z; Y�Þ; �XXÞ � �ggð �YY;�gð½X;Z�ÞÞg:

Hence, since �gg is nonsingular, �r is unique. Conversely, assume that such a

connection �r exist. Then, for every U 2 �ðM?g Þ, X, Y2 G(TM) with �gðXÞ ¼�XX;�gY ¼ �YY;

ðLUgÞðX; YÞ ¼ UgðX; YÞ � gð½U;X�;YÞ � gðX; ½U;Y�Þ

¼ U�ggð �XX; �YYÞ � �ggð�gð½U;X�Þ; �YYÞ � �ggð �XX;�gð½U; Y�Þ

¼ U�ggð �XX; �YYÞ � �ggð �rU�XX; �YYÞ � �ggð �XX; �rU

�YYÞ

¼ 0:

Thus (M, g) is a singular semi-Riemannian manifold. Ì

We shall now return to Ka€hler manifolds.


DEFINITION 4.2. Let V be a complex vector space. A function H: V � V ! Cis called a Hermitian inner product on V if

(a) H(u, v) is C-linear in v.

(b) H(u, v) = *H(v, u), where *H(v, u) is the complex conjugate of H(v, u).

The type (k, m, n) of a Hermitian inner product H on V is defined by

k = complexdim {u 2 V j H(u, v) = 0, 8v 2 V} (the complex nullity of H),

m = sup{complexdim W j W Î V and H(w, w) < 0, 80 m w 2 W} (the

complex index of H),

n = sup{complexdim W j W Î V and H(w, w) > 0, 80 m w 2 W}.

A Hermitian inner product H on V is called nondegenerate if k = 0. (V, H) is

called a Hermitian inner product space of type (k, m, n) if H is a Hermitian inner

product on V of type (k, m, n).

DEFINITION 4.3. Let E be a complex vector bundle over a manifold M. A

(smooth) H 2 G(E* � E*) is called a Hermitian metric of type (k, m, n) in Eif, for all p 2 M, (Ep, Hp) is a Hermitian inner product space of constant type

(k, m, n). (E, H) is called a Hermitian vector bundle of type (k, m, n) if H is a

Hermitian metric of type (k, m, n) in E. The complex degenerate space M?H of a

Hermitian metric H in E is defined to be the complex vector bundle

M?H ¼?p2Mfu 2 Ep j Hðu; vÞ ¼ 0;8v 2 Epg:

Remark 4.4. Let (M, J) be an almost complex manifold and let H be a

Hermitian metric in the complex vector bundle (TM, J). Then g = re(H) and � =

im(H) respectively defines a metric tensor and an almost symplectic structure on

M such that g, � and J are compatible differential geometric structures. (Con-

versely, if g, � and J are compatible differential geometric structures on a

manifold M, then H = g + i� defines a Hermitian metric in the complex vector

bundle (TM, J).) We observe that the M? is the real vector bundle underlying the

complex vector bundle M?H and TM=M? is the real vector bundle underlying

the complex vector bundle ðTM; JÞ=M?H : Thus, ðTM; �JJÞ ¼ ðTM; JÞ=M?H and the

induced (nondegenerate) Hermitian metric �HH in ðTM; JÞ=M?H can be written as�HH ¼ �ggþ i��; where �gg and �� are the induced differential geometric structures in

TM:

DEFINITION 4.5. Let g, � and J be compatible differential geometric

structures on a manifold M. Then the (nondegenerate) Hermitian metric �HH in

ðTM; �JJÞ induced by the differential geometric structures g, � and J is defined by�HH ¼ �ggþ i��; where i ¼

ffiffiffiffiffiffiffi�1p

: In addition, if (M, g) and (M, �) are, respectively,

singular semi-Riemannian and singular almost symplectic manifolds, then �HH is

called a pre-Ka€hler metric in ðTM; �JJÞ: A pre-Ka€hler metric �HH is called a Ka€hler

metric in ðTM; �JJÞ if �dd�� ¼ 0:


THEOREM 4.6. Let g, � and J be compatible differential geometric structureson a manifold M such that (M, g) and (M, �) are respectively, singular semi-Riemannian and singular almost symplectic manifolds. Let �HH ¼ �ggþ i�� be theinduced pre-Ka€hler metric in ðTM; �JJ Þ by g, � and J and let �r be the Koszulconnection in ðTM; �ggÞ. Then the following are equivalent:

(a) �r�JJ ¼ 0;(b) �r�� ¼ 0;(c) �dd �� ¼ 0 and �NN ¼ 0:

Proof. (a) () (b): Since the Koszul connection �r is a semi-Riemannian

connection in ðTM; �ggÞ; �r�gg ¼ 0: Hence, for every X 2 �ðTMÞ; �YY; �ZZ 2 �ðTMÞ;

ð �rX��Þð �YY; �ZZÞ ¼ X ��ð �YY; �ZZÞ � ��ð �rX

�YY; �ZZÞ � ��ð �YY; �rX�ZZÞ

¼ X�ggð�JJ �YY; �ZZÞ � �ggð�JJð �rX�YYÞ; �ZZÞ � �ggð�JJ �YY; �rX

�ZZÞ¼ ð �rX �ggÞð�JJ �YY; �ZZÞ þ �ggð �rXð�JJ �YYÞ; �ZZÞ � �ggð�JJð �rX

�YYÞ; �ZZÞ¼ �ggðð �rX

�JJÞ �YY; �ZZÞ:

Thus, since �gg is nondegenerate, �r�� ¼ 0 iff �r�JJ ¼ 0.

(b) () (c): We first need to prove two identities.

(I) �dd��ð �XX; �YY; �ZZÞ ¼ ð �rX��Þð �YY; �ZZÞ � ð �rY

��Þð �XX; �ZZÞ þ ð �rZ��Þð �XX; �YYÞ; where X, Y,

Z 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ:Proof of (I). Since

ð �rX��Þð �YY; �ZZÞ ¼ X ��ð �YY; �ZZÞ � ��ð �rX

�YY; �ZZÞ � ��ð �YY; �rX�ZZÞ;

ð �rY��Þð �XX; �ZZÞ ¼ Y ��ð �XX; �ZZÞ � ��ð �rY

�XX; �ZZÞ � ��ð �XX; �rY�ZZÞ;

ð �rZ��Þð �XX; �YYÞ ¼ Z ��ð �XX; �YYÞ � ��ð �rZ

�XX; �YYÞ � ��ð �XX; �rZ�YYÞ;

it follows that

ð �rX��Þð �YY; �ZZÞ � ð �rY

��Þð �XX; �ZZÞ þ ð �rZ��Þð �XX; �YYÞ

¼ X ��ð �YY; �ZZÞ � Y ��ð �XX; �ZZÞ þ Z ��ð �XX; �YYÞ þ ��ð �ZZ; �rX�YY � �rY

�XXÞ��ð �YY; �rX

�ZZ � �rZ�XXÞ þ ��ð �XX; �rY

�ZZ � �rZ�YYÞ

¼ X ��ð �YY; �ZZÞ � Y ��ð �XX; �ZZÞ þ Z ��ð �XX; �YYÞ þ ��ð �ZZ;�ð½X;Y�ÞÞ��ð �YY;�ð½X;Z�ÞÞ þ ��ð �XX;�ð½Y; Z�ÞÞ

¼ �dd��ð �XX; �YY; �ZZÞ:

(II) �dd��ð �XX; �YY; �ZZÞ � �dd��ð �XX; �JJ �YY; �JJ �ZZÞ ¼ 2�ggðð �rX�JJÞ �YY; �ZZÞ þ �ggð�JJ �XX; �NNð �YY; �ZZÞÞ, where

X 2 �ðTMÞ with �ðXÞ ¼ �XX and �YY; �ZZ 2 �ðTMÞ:


Proof of (II). Let X, Y, Z 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ:Then,

�dd��ð �XX; �YY; �ZZÞ ¼ X ��ð �YY; �ZZÞ � Y ��ð �XX; �ZZÞ þ Z ��ð �XX; �YYÞ þ ��ð �ZZ;�ð½X; Y�ÞÞ

� ��ð �YY;�ð½X; Z�ÞÞ þ ��ð �XX;�ð½Y;Z�ÞÞ

¼ X�ggð�JJ �YY; �ZZÞ � Y�ggð�JJ �XX; �ZZÞ þ Z�ggð�JJ �XX; �YYÞ þ �ggð�JJ �ZZ;�ð½X; Y�ÞÞ

� �ggð�JJ �YY;�ð½X; Z�ÞÞ þ �ggð�JJ �XX;�ð½Y;X�ÞÞ

¼ X�ggð�JJ �YY; �ZZÞ � Y�ggð�JJ �XX; �ZZÞ þ Z�ggð�JJ �XX; �YYÞ þ �ggð�JJ �ZZ; �rX�YYÞ

� �ggð�JJ �ZZ; �rY�XXÞ � �ggð�JJ �YY; �rX

�ZZÞ þ �ggð�JJ �YY; �rZ�XXÞ þ �ggð�JJ �XX;�ð½Y;Z�ÞÞ:

Similarly,

�dd��ð �XX; �JJ �YY; �JJ �ZZÞ ¼ �ggð�JJ �YY; �rX�ZZÞ � Z�ggð �XX; �JJ �YYÞ þ �ggð �rZ

�XX; �JJ �YYÞ � �ggð �XX;�ð½JY;Z�ÞÞ

þX�ggð�JJ �ZZ; �YYÞ � �ggð�JJ �ZZ; �rX�YYÞ þ Y�ggð �XX; �JJ �ZZÞ �ggð �rY

�XX; �JJ �ZZÞ

þ �ggð �XX;�ð½JZ; Y�ÞÞ þ �ggð�JJ �XX; �ð½JY; JZ�ÞÞ:

Then, since �ggð�JJ �XX; �JJ �YYÞ ¼ �ggð �XX; �YYÞ for every �XX; �YY 2 �ðTMÞ;�dd��ð �XX; �YY; �ZZÞ � �dd��ð �XX; �JJ �YY; �JJ �ZZÞ ¼ 2X�ggð�JJ �YY; �ZZÞ � 2�ggð�JJ �YY; �rX

�ZZÞ þ 2�ggð�JJ �ZZ; �rX�YYÞ

þ �ggð�JJ �XX;�ð½Y;Z� þ J½JY;Z� þ J½Y; JZ�

� ½JY; JZ�ÞÞ

¼ 2�ggðð �rX�JJÞ �YY; �ZZÞ þ �ggð�JJ �XX; �NNð �YY; �ZZÞÞ:

We now complete the proof of the theorem.

(a) + (b) Á (c) If �r�� ¼ 0, then from identity (I), �dd�� ¼ 0. Also since �r�JJ ¼ 0

from identity (II), �ggð�JJ �XX; �NNð �YY �ZZÞÞ ¼ 0 for every �XX; �YY; �ZZ 2 �ðTMÞ: Hence �NN ¼ 0.

(c) Á (b) If �dd�� ¼ 0 and �NN ¼ 0, then from identity (II), �ggðð �rX�JJÞ �YY; �ZZÞ ¼ 0 for

every X 2 �ðTMÞ and �YY; �ZZ 2 �ðTMÞ: Hence �r�JJ ¼ 0: Ì

DEFINITION 4.7. Let g be a metric tensor of type (2k, 2m, 2n), � be an almost

symplectic structure and J be an almost complex structure on a manifold M. If g,

� and J are compatible differential geometric structures on M and, (M, g) and

(M, �) are respectively, singular semi-Riemannian and singular almost

symplectic manifolds then (M, g, �, J) is called a pre-Ka€hler manifold of type

(k, m, n). A pre Ka€hler manifold (M, g, �, J) of type (k, m, n) is called a Ka€hler

manifold of type (k, m, n) if �dd �� ¼ 0 and �NN ¼ 0:

Remark 4.8. Note that if (M, g, �, J) is a Ka€hler manifold of type (k U 1, m, n)

then J is not necessarily integrable since �NN ¼ 0 does not imply that N = 0. For


example, let (M1, J1) be an almost complex manifold such that J1 is not the

canonical almost complex structure of some complex manifold structure on M1,

and let (M2, g2, �2, J2) be a Ka€hler manifold of type (0, m, n). (Hence, M2 is a

complex manifold with canonical almost complex structure J2.) Let P1: M1 �M2

Y M1 and P2: M1 � M2 Y M2 be the canonical projections. Then (M1 � M2,

P2 * g2) and (M1 � M2, P2 * �2) are respectively singular semi-Riemannian and

symplectic manifolds with nullity k = dimM1. Define an almost complex structure

J on M1 �M1 such that, for x 2 Tðp;qÞðM1 �M2Þ; Jx ¼ ðP1* jTpM1�0Þ�1J1ðP1* xÞþ

ðP2 * j0�TqM2Þ�1J2ðP2 * xÞ: Hence, P2*g2, P2*� and J are compatible on M1 � M2

but since M1 is not a complex manifold, J is not the canonical almost complex

structure of some complex manifold structure on M1 � M2. However, notice that

(M1 � M2, P2*g2, P2*�, J) is a Ka€hler manifold of type (k / 2, m, n), where

k = dimM1.

COROLLARY 4.9. Let (M, g, �, �JJ) be a Ka€hler manifold of type (k, m, n). Thenthe Koszul connection �r can be extended to a connection in the complex bundleðTM; �JJÞ. In this case, �r �HH ¼ 0:

Proof. Note that ðTM; �JJÞ and the underlying real vector bundle have the same

fibers. Hence, if �XX 2 �ðTMÞ and f þ ih 2 C1ðM;CÞ;�rYð f þ ihÞ �XX ¼ �rYð f �XX þ h �JJ �XXÞ

¼ ðYf Þ �XX þ f �rY�XX þ ðYhÞ�JJ �XX þ h�JJð �rY

�XXÞ¼ ððYf Þ þ iðYhÞÞ �XX þ ð f þ ihÞ �rY

�XX

¼ ðYð f þ ihÞÞ �XX þ ðf þ ihÞ �rY�XX

for every Y 2 G(TM). Hence, �r is also a connection in ðTM; �JJÞ. Thus, �r �HH ¼�r�ggþ i �r�� ¼ 0 from Theorem 4.6. Ì

COROLLARY 4.10. Let (M, g) be a singular semi-Riemannian and (M, J)

be an almost complex manifold such that g(JX, JY) = g(X, Y) and LUJ 2�ðHomðTM; M?g ÞÞ for every X;Y 2 �ðTMÞ;U 2 �ðM?g Þ: Then there exists aunique almost symplectic structure � on M such that (M, g, �, J) is a pre-Ka€hlermanifold.

Proof. Immediate from Proposition 2.11 and Corollary 2.10. Ì

COROLLARY 4.11. Let (M, g) be a singular almost symplectic and (M, J) bean almost complex manifold such that �(JX, JY) = �(X, Y) and LUJ 2 �ðHom

ðTM;M?�ÞÞ for every X;Y 2 �ðTMÞ;U 2 �ðM?�Þ. Then there exists a uniquemetric tensor g on M such that (M, g, �, J) is a pre-Ka€hler manifold.

Proof. Immediate from Proposition 2.12 and Corollary 2.10. Ì

Remark 4.12. In Corollary 4.9, if g is a metric tensor of type (2k, 2m, 2n), then

(M, g, �, J) is a pre-Ka€hler manifold of type (k, m, n). However, in Corollary

4.10, if � is an almost symplectic structure with nullity 2k then we can only say


that (M, g, �, J) is a pre-Ka€hler manifold with nullity k unless we know Jexplicitly.

5. Curvature of Degenerate Ka€hler Manifolds

We recall from [2] that the curvature tensor �RR of the Koszul connection �r in

ðTMg; �ggÞ of a singular semi-Riemannian manifold (M, g) satisfies the following

identities (cf. Theorem 4.9. of [2]),

(a) �RRðX; YÞ ¼ � �RRðY;XÞ(b) �ggð �RRðX; YÞ �ZZ; �VVÞ ¼ ��ggð �RRðX;YÞ �VV; �ZZÞ(c) d �r �RR ¼ 0, where d �r is the exterior covariant derivative operator with respect

to �r (second Bianchi identity)

(d) �RRðX; YÞ �ZZ þ �RRðY;ZÞ �XX þ �RRðZ;XÞ �YY ¼ 0 (first Bianchi identity)

(e) �ggð �RRðX; YÞ �ZZ; �VVÞ ¼ �ggð �RRðZ;VÞ �XX; �YYÞ, where X; Y; Z;V 2 �ðTMÞ with �gðXÞ ¼�XX;�gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ;�gðVÞ ¼ �VV.

The intrinsic curvature tensor RR of ðTM; �ggÞ is defined by RRð �XX; �YYÞ �ZZ ¼ �RRðX;YÞ �ZZ, where X; Y 2 �ðTMÞ with �gðXÞ ¼ �XX;�ðYÞ ¼ �YY and �ZZ 2 �ðTMÞ (cf.

Definition 4.14 of [2]). It is easy to see that RR satisfies the above identities of �RRexcept ðcÞ since d �rRR is not defined. However, if we consider RR as a covariant 3-

tensor in TMg with values in TMg then RR satisfies the following second Bianchi

identity.

THEOREM 5.1. Let (M, g) be a singular semi-Riemannian manifold. Then theintrinsic curvature tensor RR of ðTMg; �ggÞ satisfies the second Bianchi identity

ð �rXRRÞð �YY; �ZZ; �VVÞ þ ð �rYRRÞð �ZZ; �XX; �VVÞ þ ð �rZRRÞð �XX; �YY; �VVÞ ¼ 0;

where X, Y, Z 2 G(TM) with �gðXÞ ¼ �XX;�gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ.

Proof. Since d �r �RR ¼ 0,

ð �rXRRÞð �YY; �ZZ; �VVÞ þ ð �rYRRÞð �ZZ; �XX; �VVÞ þ ð �rrZRRÞð �XX; �YY; �VVÞ

¼ �rXRRð �YY; �ZZÞ �VV� RRð �rX�YY; �ZZÞ �VV � RRð �YY; �rX

�ZZÞ �VV � RRð �YY; �ZZÞ �rX�VV þ �rYRRð �ZZ; �XXÞ �VV

� RRð �rY�ZZ; �XXÞ �VV � RRð �ZZ; �rY

�XXÞ �VV � RRð �ZZ; �XXÞ�rY�VV þ �rZRRð �XX; �YYÞ �VV

� RRð �rZ�XX; �YYÞ �VV� RRð �XX; �rZ

�YYÞ �VV � RRð �XX; �YYÞ �rZ�VV

¼ �rXRRð �YY; �ZZÞ �VV þ �rYRRð �ZZ; �XXÞ �VV þ �rZRRð �XX; �YYÞ �VV � RRð�ð½X;Y�Þ; �ZZÞ �VV

� RRð�ð½Y; ZÞ; �XXÞ �VV þ RRð�ð½X;Z�Þ; �YYÞ �VV � RRð �YY; �ZZÞ�rX�VV � RRð �ZZ; �XXÞ �rY

�VV

� RRð �XX; �YYÞ �rZ�VV


¼ �rX�RRð �YY; �ZZÞ �VV þ �rY

�RRðZ;XÞ �VVþ �rZ�RRðX; YÞ �VV� �RRð½X; Y�;ZÞ �VV� �RRð½Y; Z�;XÞ �VV

þ �RRð½X;Z�;YÞ �VV � �RRðY; ZÞ �rX�VV � �RRðZ;XÞ �rY

�VV � �RRðX;YÞ �rZ�VV

¼ ðd �r �RRÞðX;Y;ZÞð �VVÞ

¼ 0:Ì

Let x; y 2 ðTMgÞp be linearly independent vectors. Then, �PP ¼ spanf�xx; �yyg is

called a plane in TMg. Let �QQð�xx; �yyÞ ¼ �ggð�xx; �xxÞ�ggð�yy; �yyÞ � ½�ggð�xx; �yy�2, where �xx; �yy 2ðTMgÞp. A plane �PP ¼ spanf�xx; �yyg in TMg is called nondegenerate if �QQð�xx; �yyÞ 6¼ 0

and called degenerate if �QQð�xx; �yyÞ ¼ 0. If (M, g) is a singular semi-Riemannian

manifold, then the curvature of a nondegenerate plane �PP ¼ spanf�xx; �yyg is defined

by �KKð �PPÞ ¼ �ggðRRð�xx; �yyÞ�yy; �xxÞ= �QQð�xx; �yyÞ. (see Section 4 of [2]). A singular semi-

Riemannian manifold (M, g) is said to have constant curvature C at P 2 M if the

curvature of every nondegenerate plane in ðTMgÞp is equal to C. In this case,

RRð�xx; �yyÞ�zz ¼ Cð�ggð�zz; �yyÞ�xx� �ggð�zz; �xxÞ�yyÞ for every �xx; �yy; �zz 2 ðTMgÞp (cf. Proposition 4.13

of [2]).

We shall first prove the Schur Lemma for singular semi-Riemannian

manifolds.

THEOREM 5.2. Let (M, g) be a connected singular semi-Riemannian manifold.If (M, g) has constant curvature at each point and rank ðTMgÞ U 3, then (M, g) isof constant curvature, that is, curvature of every nondegenerate plane in TMg isequal to a constant.

Proof. Let R0R0ð�xx; �yyÞ�zz ¼ �ggð�zz; �yyÞ�xx� �ggð�zz; �xxÞ�yy for every �xx; �yy; �zz 2 TMg. Thus RR ¼f RR0, where f is a smooth function on M. Then, since �rRR0 ¼ 0, for every X, Y, Z,

V 2 G(TM) with �gðXÞ ¼ �XX;�gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ;�gðVÞ ¼ �VV,

ðXf ÞRR0ð �YY; �ZZ; �VVÞ þ ðYf ÞRR0ð �ZZ; �XX; �VVÞ þ ðZf ÞRR0ð �XX; �YY; �VVÞ ¼ 0

from the second Bianchi identity. Thus,

ðXf Þð�ggð �VV; �ZZÞ �YY� �ggð �VV; �YYÞ �ZZÞ þ ðYf Þð�ggð �VV; �XXÞ �ZZ� �ggð �VV; �ZZÞ �XXÞ þ

þ ðZf Þð�ggð �VV; �YYÞ �XX� �ggð �VV; �XXÞ �YYÞ ¼ 0:

For an arbitrary X 2 �ðM?g Þ, choosing nondegenerate nonnull X, Y, Z, V 2 G(TM)

such that V = Z and �gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ are orthonormal in ðTMg; �ggÞ, we

obtain �ggð �ZZ; �ZZÞXðf Þ �YY ¼ 0. Hence, X( f ) = 0 for every X 2 M?g . For an arbitrary

nondegenerate X 2 G(TM) with jgðx; xÞj ¼ 1; choosing nondegenerate nonnull Y,

Z, V 2 G(TM) with Z = V and �gðXÞ ¼ �XX;�gðYÞ ¼ �YY;�gðZÞ ¼ �ZZ are ortho-

normal in ðTMg; �ggÞ; we obtain �ggð �ZZ; �ZZÞðXf Þ �YY � �ggð �ZZ; �ZZÞðYf Þ �XX ¼ 0. Hence Xf = 0

for every nondegenerate nonnull X 2 G(TM). Hence, it follows that Xf = 0 for


every X 2 G(TM) since any null vector can be written as a linear combination of

nondegenerate nonnull vectors. Ì

DEFINITION 5.3. Let (M, g) be a singular semi-Riemannian manifold with

metric g of type (k, m, n). The intrinsic Ricci curvature cRicRic and the intrinsic

scalar curvature SS of ðTM; �ggÞ; is defined by

cRicRicð �XX; �YYÞ ¼X

i

�i �ggðRRð �EEi; �XXÞ �YY; �EEiÞ

and SS ¼P

i�icRicRicð �EEi; �EEiÞ, where { �EEi} is a local semi-Riemannian basis in ðTM; �ggÞ

and �i ¼ �ggð �EEi; �EEiÞ:

Remark 5.4. Note that SS ¼P

i6¼j�KKð �PPijÞ, where �PPij ¼ spanf �EEi; �EEjg:

We shall now return to the curvature of Ka€hler manifolds.

THEOREM 5.5. Let (M, g, �, J) be a Ka€hler manifold of type (k, m, n). Then forall �XX; �YY; �ZZ; �VV 2 �ðTMÞ;(a) RRð �XX; �YYÞ � �JJ ¼ �JJ � RRð �XX; �YYÞ and RRð�JJ �XX; �JJ �YYÞ ¼ RRð �XX; �YYÞ(b) �HHðRRð �XX; �YYÞ �ZZ; �VVÞ ¼ � �HHðRRð �XX; �YYÞ �VV; �ZZÞ(c) cRicRicð�JJ �XX; �JJ �YYÞ ¼ cRicRicð �XX; �YYÞ ¼ 1=2tr �JJ � RRð �XX; �JJ �YYÞ:

Proof. Let X, Y, Z, V 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ;�ðVÞ ¼ �VV.

(a) RRð �XX; �YYÞ�JJ �ZZ ¼ �RRðX; YÞ�JJ �ZZ

¼ �rX�rY

�JJ �ZZ � �rY�rX

�JJ �ZZ � �r½X;Y� �JJ �ZZ

¼ �JJð �rX�rY

�ZZ � �rY�rX

�ZZ � �r½X;Y� �ZZÞ¼ �JJð �RRðX; YÞ �ZZÞ¼ �JJðRRð �XX; �YYÞ �ZZÞ

since �r�JJ ¼ 0. For RRð�JJ �XX; �JJ �YYÞ ¼ RRð �XX; �YYÞ; since

�ggðRRð�JJ �XX; �JJ �YYÞ �ZZ; �VVÞ ¼ �ggðRRð �ZZ; �VVÞ�JJ �XX; �JJ �YYÞ¼ �ggð�JJRRð �ZZ; �VVÞ �XX; �JJ �YYÞ¼ �ggðRRð �ZZ; �VVÞ �XX; �YYÞ¼ �ggðRRð �XX; �YYÞ �ZZ; �VVÞ

for every �ZZ; �VV 2 �ðTMÞ, it follows that RRð�JJ �XX; �JJ �YYÞ ¼ RRð �XX; �YYÞ:


(b) Since �r �HH ¼ 0;

�HHðRRð �XX; �YYÞ; �VVÞ ¼ �HHð �RRðX;YÞ �ZZ; �VVÞ¼ �HHð �rX

�rY�ZZ � �rY

�rX�ZZ � �r½X;Y� �ZZ; �VVÞ

¼ X �HHð �rY�ZZ; �VVÞ � �HHð �rY

�ZZ; �rX�VVÞ � Y �HHð �rX

�ZZ; �VVÞ þþ �HHð �rX

�ZZ; �rY�VVÞ � ½X;Y� �HHð �ZZ; �VVÞ þ �HHð �ZZ; �r½X;Y� �VVÞ

¼ �HHð �ZZ; �rY�rX

�VVÞ � �HHð �ZZ; �rX�rY

�VV þ �HHð �ZZ; �r½X;Y� �VVÞ¼ � �HHð �ZZ; �RRð �XX; �YYÞ �VVÞ¼ � �HHðRRð �XX; �YYÞ �VV; �ZZÞ:

(c) Let { �EEi} be a local semi-Riemannian basis in ðTMÞ; �ggÞ. Then, f�JJ �EEig is

also a local semi-Riemannian basis in ðTM; �ggÞ such that �ggð �EEi; �EEiÞ ¼�ggð�JJ �EEi; �JJ �EEiÞ ¼ �i: Thus,

cRicRicð�JJ �XX; �JJ �YYÞ ¼X

i

�i �ggðRRð �EEi; �JJ �XXÞ�JJ �YY; �EEiÞ

¼X

i

�i �ggðRRð�JJ �EEi; �XXÞ �YY; �JJ �EEiÞ

¼ cRicRicð �XX; �YYÞ:

The proof of cRicRicð �XX; �YYÞ ¼ 1=2tr�JJ � RRð �XX; �JJ �YYÞ is the duplicate of its standard

proof for nondegenerate Ka€hler manifolds, for example, see either p. 149

of [1] or p. 271 of [4]. Ì

DEFINITION 5.6. Let [M, g, �, J) be a pre-Ka€hler manifold. A plane �PP in ðTMÞis called holomorphic if it is invariant under �JJ.

Remark 5.7. Note that if (M, g, �, J) is a pre-Ka€hler manifold then the

restriction of �gg to a holomorphic plane �PP in TM has signature either (j, j) or (0,

0) or (+, +). Thus, if �PP is a nondegenerate plane then the curvature �KKð �PPÞ ¼�ggðRRð�xx; �JJ�xxÞ�JJ�xx; �xxÞ, where �xx 2 �PP is a unit vector.

Note that if (M, g) is a singular semi-Riemannian manifold then the covariant

4-tensor F in ðTMg; �ggÞ which is defined by Fð�xx; �yy; �zz; �vvÞ ¼ �ggðRRð�xx; �yyÞ�zz; �vvÞ satisfies

the identities

(a) Fð�xx; �yy; �zz; �vvÞ ¼ �Fð�yy; �xx; �zz; �vvÞ;(b) Fð�xx; �yy; �zz; �vvÞ ¼ �Fð�xx; �yy; �vv; �zzÞ;(c) Fð�xx; �yy; �zz; �vvÞ þ Fð�yy; �zz; �xx; �vvÞ þ Fð�zz; �xx; �yy; �vvÞ ¼ 0;(d) Fð�xx; �yy; �zz; �vvÞ ¼ Fð�zz; �vv; �xx; �yyÞ;

for every �xx; �yy; �zz; �vv 2 TMg. More generally, an arbitrary covariant 4-tensor field Fin TMg is called curvature-like if it satisfies the above identities.


DEFINITION 5.8. Let (M, g, �, J) be a pre-Ka€hler manifold. A curvature-like

tensor field F in TM is called holomorphic curvature-like if

Fð�JJ�xx; �JJ�yy; �zz; �vvÞ ¼ Fð�xx; �yy; �zz; �vvÞ ¼ Fð�xx; �yy; �JJ�zz; �JJ�vvÞ

for every �xx; �yy; �zz; �vv 2 TM.

LEMMA 5.9. Let (M, g, �, J) be a pre-Ka€hler manifold. If a holomorphiccurvature-like tensor field in TM satisfies Fð�xx; �JJ�xx; �JJ�xx; �xxÞ ¼ 0 for every �xx 2 TMp

then Fp ¼ 0.Proof. See p. 166 of [1]. Ì

PROPOSITION 5.10. Let (M, g, �, J) be a Ka€hler manifold. Then

(a) RRð�xx; �yy; �zz; �vvÞ ¼ �ggðRRð�xx; �yyÞ�zzÞ; �vvÞ is a holomorphic curvature-like tensor field inTM:

(b) RR1ð�xx; �yy; �zz; �vvÞ ¼1=4f�ggð�zz; �yyÞ�ggð�xx; �vvÞ � �ggð�zz; �xxÞ�ggð�yy; �vvÞ þ �ggð�JJ�yy; �zzÞ�ggð�JJ�xx; �vvÞ � �ggð�JJ�xx; �zzÞ�ggð�JJ�yy; �vvÞ þ 2�ggð�xx; �JJ�yyÞ�ggð�JJ�zz; �vvÞg is a parallel holomorphic curvature-like ten-sor field in TM: In particular, RR1ð�xx; �yy; �yy; �xxÞ ¼ 1=4f �QQð�xx; �yyÞ þ 3½�ggð�xx; �JJ�yy�2g andRR1ð�xx; �JJ�xx; �JJ�xx; �xxÞ ¼ ½�ggð�xx; �xxÞ�2 where �QQð�xx; �yyÞ ¼ �ggð�xx; �xxÞ�ggð�yy; �yyÞ � ½�ggð�xx; �yyÞ�2.

Proof. (a) It follows from the definition of RR and Theorem 5.5 that RR is a

holomorphic curvature-like tensor field in TM.(b) It is straightforward to check that RR1 is a holomorphic curvature-like

tensorfield in TM. Since �JJ and �gg are parallel, RR1 is also parallel. Ì

THEOREM 5.11. Let (M, g, �, J) be a Ka€hler manifold. If �KKð �PPÞ ¼ C for everynondegenerate holomorphic plane �PP in TMp then RR ¼ CRR1 at p 2 M.

Proof. Define F ¼ RR� CRR1: Then F is a holomorphic curvature-like tensor in

TM such that Fð�xx; �JJ�xx; �JJ�xx; �xxÞ ¼ 0 for every nonnull �xx 2 TMp: Hence, from con-

tinuity, Fð�xx; �JJ�xx; �JJ�xx; �xxÞ ¼ 0 for every null �xx 2 TMp: Then from Lemma 5.9, RR ¼CRR1: Ì

THEOREM 5.12. Let (M, g, �, J) be a connected Ka€hler manifold. If rankðTMÞ U 4 and curvatures of the nondegenerate holomorphic planes in TMp areconstant at each point, then (M, g, �, J) is of constant holomorphic curvature,i.e. curvatures of the nondegenerate holomorphic planes in TM are equal to aconstant.

Proof. Let RR1ð�xx; �yy; �zzÞ¼1=4f�ggð�zz; �yyÞ�xx� �ggð�zz; �xxÞ�yyþ �ggð�JJ�yy; �zzÞ�JJ�xx� �ggð�JJ�xx; �zzÞ�JJ�yyþ 2�ggð�xx; �JJ�yyÞ�JJ�zzg for every �xx; �yy; �zz 2 �ðTMÞ: Hence RRð �XX; �YY; �ZZÞ ¼ f RR1ð �XX; �YY; �ZZÞ for every�XX; �YY; �ZZ 2 �ðTMÞ; where f is a smooth function on M. Then, from the second

Bianchi identity,

ðXf ÞRR1ð �YY; �ZZ; �VVÞ þ ðYf ÞRR1ð �ZZ; �XX; �VVÞ þ ðZf ÞRR1ð �XX; �YY; �VVÞ ¼ 0

where X, Y, Z 2 G(TM) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ and �VV 2 �ðTMÞ.


Thus

ðXf Þ½�ggð �VV; �ZZÞ �YY � �ggð �VV; �YYÞ �ZZ þ �ggð �VV; �JJ �ZZÞ�JJ �YY � �ggð �VV; �JJ �YYÞ�JJ �ZZ þ 2�ggð �YY; �JJ �ZZÞ�JJ �VV�

þ ðYf Þ½�ggð �VV; �XXÞ �ZZ � �ggð �VV; �ZZÞ �XX þ �ggð �VV; �JJ �XXÞ �JJ �ZZ � �ggð �VV; �JJ �ZZÞ �JJ �XX

þ 2�ggð �ZZ; �JJ �XXÞ �JJ �VV� þ ðZ f Þ½�ggð �VV; �YYÞ �XX � �ggð �VV; �XXÞ �YY þ �ggð �VV; �JJ �YYÞ�JJ �XXÞ

� �ggð �VV; �JJ �XXÞ�JJ �YY þ 2gð �XX; �JJ �YYÞ�JJ �VV�

¼ 0:

For an arbitrary X 2 �ðM?Þ, choosing nondegenerate nonnull Y, Z, V 2G(TM) with V = Z and �ðYÞ ¼ �YY;�ðZÞ ¼ �ZZ; �JJ �YY are orthonormal in TM, we

obtain ðXf Þ�ggð �ZZ; �ZZÞ �YY ¼ 0: Hence Xf = 0 for every X 2 �ðM?Þ:For an arbitrary (nondegenerate) X 2 G(TM) with |g(X, X)| = 1, choosing

nondegenerate nonnull Y, Z, V 2 G(TM) with V = Z and �ðXÞ ¼ �XX; �JJ �XX;�ðYÞ ¼�YY; �ZZ ¼ �JJ �YY are orthonormal, we obtain

4ðXf Þ�ggð �ZZ; �ZZÞ �YY � ðYf Þ�ggð �ZZ; �ZZÞ �XX þ ðZf Þ�ggð �YY; �YYÞ�JJ �XX ¼ 0:

Hence Xf = 0 for every nondegenerate nonnull X 2 G(TM). Thus, as in Theorem

5.2, Xf = 0 for every X 2 G(TM). Ì

PROPOSITION 5.13. Let (M, g, �, J) be a Ka€hler manifold of constantholomorphic curvature C with rankðTMÞ U 4. Then,

(a) If (M, g, �, J) is of constant curvature, then C = 0.

(b) If indexð�ggÞ ¼ 0 then curvatures of planes �PP in TM satisfy 14

� �C r �KKð �PPÞ r C

for C > 0 and C r �KKð �PPÞ r ð1=4ÞC for C < 0.

(c) If 0 < indexð�ggÞ < rankTM then the curvatures of the nondegenerate planesin TM cannot be bound either from above or below unless (M, g) is flat.

Proof. Since (M, g, �, J) is of constant holomorphic curvature C, RRð�xx; �yyÞ�zz ¼CRR1ð�xx; �yy; �zzÞ for every �xx; �yy; �zz 2 TM: Hence, if �PP ¼ spanf�xx; �yyg is a nondegenerate

plane in TM then

�KKð �PPÞ ¼ �ggðRRð�xx; �yyÞ�yy; �xxÞ= �QQð�xx; �yyÞ¼ CRR1ð�xx; �yy; �yy; �xxÞ= �QQð�xx; �yyÞ

¼ 1

4

� �Cf1þ 3½�ggð�xx; �JJ�yyÞ�2= �QQð�xx; �yyÞg:

(a) Let �PP be a nondegenerate plane in TM with �PP ? �JJð �PPÞ: Then �KKð �PPÞ ¼14

� ��KKð�JJð �PPÞÞ and hence C ¼ �KKð �PPÞ ¼ 0:

(b) Since indexð�ggÞ ¼ 0; for every orthonormal �xx; �yy 2 �PP; we have �QQð�xx; �yyÞ ¼ 1

and j�ggð�xx; �JJ�yyÞj r 1: Thus, 14

� �C r �KKð �PPÞ r C for C > 0 and C r �KKð �PPÞ r 1

4

� �

C for C < 0:


(c) We note that the proof of Theorem 1 in [3] also valid for ðTM; �ggÞ. Hence,

(M, g) is of constant curvature if the curvatures of nondegenerate planes are

bounded either from above or below. Then, from (a), (M, g) is flat. Ì

6. Complex Degenerate Ka€hler Manifolds

In Section 4, we only considered Ka€hler structure on almost complex manifolds.

In this section we shall study the Ka€hler structure on complex manifolds. Let Mbe a complex manifold with canonical almost complex structure J. We shall

denote the holomorphic tangent space of M at p 2 M by (TpM, Jp) and the

holomorphic tangent bundle by (TM, J). We denote the complexification of the

real tangent bundle of a complex manifold M by TMC. Also, let TM(1,0) and

TM(0,1) be the eigenbundles corresponding to the eigenvalues i and ji of the

complex linear extension of the canonical almost complex structure J of M to

TMC. We shall also think of TM(1,0) as the holomorphic tangent bundle of a

complex manifold M via the isomorphism between the complex vector bundles

(TM, J) and TM(1,0) ( see pp. 264Y268 of [4] for the notation and terminology of

this section).

DEFINITION 6.1. Let H be a Hermitian metric on a complex manifold M with

canonical almost complex structure J. The associated nondegenerate holomor-

phic tangent bundle ðTM; �JJÞ of M is defined by ðTM; �JJÞ ¼ ðTM; JÞ=M?H and �:ðTM; JÞ ! ðTM; �JJÞ is the canonical projection.

The induced (nondegenerate) Hermitian metric �HH in ðTM; �JJÞ is defined by�HHð �XX; �YYÞ ¼ HðX; YÞ, where X, Y 2 G(TM, J) with �ðXÞ ¼ �XX;�ðYÞ ¼ �YY.

Note that ðTM; �JJÞ is well-defined and the almost complex structure �JJ in TM is

induced from the canonical almost complex structure J of M as in Definition 3.1

(see the Remark 4.4). Also, since the real vector bundle M? underlying M?H is of

constant rank and invariant under the canonical almost complex structure J of M,

M?H is a holomorphic vector bundle over M.

We recall that a section X of a holomorphic vector bundle E over a complex

manifold M is called holomorphic if X: M Y E is a holomorphic map between

complex manifolds M and E.

LEMMA 6.2. Let H be a Hermitian metric on a complex manifold M and letðTM; �JJÞ be the associated nondegenerate holomorphic tangent bundle of M. Thenthere exist a unique connection l in ðTM; �JJÞ such that, when l extended com-plex linearly in the first argument,

(a) v �HHð �XX; �YYÞ ¼ �HHðrv*�XX; �YYÞ þ �HHð �XX;rv

�YYÞ; where �XX; �YY 2 �ðTMÞ; v 2 TMC and

*v is the complex conjugate of v.


(b) For every holomorphic �XX 2 �ðTMÞ;r �XX 2 �ðHomðTMC;TMÞÞ is of type (1,

0), that is, rv�XX ¼ 0 for all v 2 TMð0;1Þ:

Proof. See Theorem 8.52 of [4] on p. 268. (The proof of this theorem only

involves the nondegeneracy of the Hermitian metric. Therefore, it is also valid

for nondegenerate indefinite Hermitian metrics.)

We shall show that if (M, g, �, J) is a pre-Ka€hler manifold, where J is the

canonical almost complex structure of a complex manifold M, then (M, g, �, J) is

a Ka€hler manifold iff the Koszul connection �r is a Hermitian connection of type

(1, 0) in TM when extended complex linearly in the first argument. Ì

DEFINITION 6.3. Let (M, g, �, J) be a pre-Ka€hler manifold, where J is the

canonical almost complex structure on a complex manifold M. The Lie deriv-

ative �LLX�JJ of the almost complex structure �JJ of TM along X 2 �ðTMÞ is defined

by ð�LLX�JJÞ �YY ¼ �ðLXJÞYÞ, where Y 2 �ðTMÞ with �ðYÞ ¼ �YY.

To show that �LLX�JJ is well defined, let Y1;Y2 2 �ðTMÞ with �ðY1Þ ¼ �YY ¼

�ðY2Þ: Thus, Y2 ¼ Y1 þ U; where U 2 �ðM?Þ: Then,

�ððLXJÞY2Þ ¼ �ððLXJÞðY1 þ UÞÞ ¼ �ððLXJÞY1 þ ðLXJÞUÞ:But since the Nijenhuis tensor N of J is vanishing, [JX, JU] = [X, U] + J[JX, U] +

J[X, JU] and, therefore,

�ððLXJÞUÞ ¼ �ð½X; JU� � J½X;U�Þ¼ ��ðJ½JX; JU� þ ½X; JU�Þ¼ �ðJððLJUJÞXÞÞ¼ 0;

since LUJ 2 �ðHomðTM;M?ÞÞ and M? is invariant under J. Hence, �ððLXJÞY2Þ ¼ �ððLXJÞY1Þ:

LEMMA 6.4. Let (M, g, �, J) be a Ka€hler manifold, where J is the canonicalalmost complex structure on a complex manifold M. Then, for X 2 G(TM) with�ðXÞ ¼ �XX;LX

�JJ ¼ 0 iff �JJð �rY�XXÞ ¼ �rJY

�XX for every Y 2 G(TM), where �r is theKoszul connection.

Proof. Let Y 2 GTM with �ðYÞ ¼ �YY. Then, since �r�JJ ¼ 0;

ðLX�JJÞ�YY ¼ �ð½X; JY� � J½X;Y�Þ¼ �ð½X; JY�Þ � �JJð�ð½X;Y�ÞÞ¼ �rXð�JJ �YYÞ � �rJY

�XX � �JJð �rX�YY � �rY

�XXÞ¼ �JJð �rY

�XXÞ � �rJY�XX:

Thus �LLX�JJ ¼ 0 iff �JJð �rY

�XXÞ ¼ �rJY�XX for every Y 2 G(TM). Ì


LEMMA 6.5. Let (M, g, �, J) be a Ka€hler manifold, where J is the canonicalalmost complex structure of a complex manifold M. Then,

(a) If X 2 G(TM) is holomorphic, then �ðXÞ ¼ �XX 2 �ðTMÞ is holomorphic. Inparticular, �LLXþU

�JJ ¼ 0 for every U 2 G(TM) and holomorphic X 2 G(TM).

(b) If �XX 2 �ðTMÞ is holomorphic, then there exist a holomorphic X 2 G(TM)

with �ðXÞ ¼ �XX.

(c) If �XX 2 �ðTMÞ is holomorphic, then �JJ �XX 2 �ðTMÞ is holomorphic.

Proof. (a) Notice that �: TM! TM is a holomorphic map between the

complex manifolds TM and TM, since �* � J* ¼ �JJ* ��* by definition; where J*and J* are the canonical almost complex structures of the complex manifolds TM

and TM respectively. Thus, if X 2 G(TM) is holomorphic, then � � X 2 �ðTMÞ is

holomorphic. Also, if X 2 G(TM) is holomorphic, then LXJ ¼ 0 (cf. Corollary

8.39 of [4]) and, hence, �LLX�JJ ¼ 0 by definition. Thus, since �LLU

�JJ ¼ 0 for every

U 2 �ðM?Þ(cf. Corollary 2.10), �LLXþU�JJ ¼ 0 for every U 2 �ðM?Þ and holomor-

phic X 2 G(TM).

(b) First we shall construct a holomorphic subbundle of TM transversal to M?.

Let h be a Riemannian metric on M such that h(JX, JY) = h(X, Y) for every X, Y 2G(TM) such Riemannian metrics exist (see Theorem 8.13 of [4]) and let E be a

vector bundle orthogonal to M? with respect to Riemannian metric h. Then, for Y2 G(E), h(X, Y) = h(JX, JY) = 0 for every X 2 �ðM?Þ. But since J is nonsingular

and M? is invariant under J, JY 2 G(E). Thus E is invariant under J and

therefore, E is a holomorphic vector bundle over M. Then, since �jE: E! TM is

a bundle isomorphism with �JJ ��jE ¼ �jE � J; �jE is a 1Y1 holomorphic map

between the complex manifolds E and TM: Hence, since X ¼ ð�jEÞ�1 �XX is a

holomorphic section of E for every holomorphic section �XX of TM, X is also a

holomorphic section of TM.

(c) Let X 2 G(TM) be holomorphic with �ðXÞ¼ �XX: Then, since JX is also

holomorphic (cf. Corollary 8.38 of [4]), �ðJXÞ¼ �JJ �XX 2 �ðTMÞ is holomorphic. Ì

COROLLARY 6.6. Let (M, g, �, J) be a Ka€hler manifold, where J is thecanonical almost complex structure of a complex manifold M. If �XX 2 �ðTMÞ isholomorphic, then �JJð �rY

�XXÞ ¼ �rJY�XX for every Y 2 G(TM), where �r is the Koszul

connection.

Proof. If �XX 2 �ðTMÞ is holomorphic then there exist a holomorphic X 2G(TM) with �ðXÞ ¼ �XX such that �LLX

�JJ ¼ 0 (cf. Lemma 6.5). Then, from Lemma

6.4, �JJð �rY�XXÞ ¼ �rJY

�XX for every Y 2 G(TM). Ì

THEOREM 6.7. Let (M, g, �, J) be a pre-Ka€hler manifold, where J is thecanonical almost complex structure of a complex manifold M. Then (M, g, �, J)

is a Ka€hler manifold iff the unique Hermitian connection of type (1, 0) in ðTM; �JJÞis the extension of the Koszul connection �r complex linearly in the firstargument.


Proof. Assume that (M, g, �, J) is a Ka€hler manifold. We show that the

extension of the Koszul connection satisfies the assumptions (a) and (b) of

Lemma 6.2 (recall that the Koszul connection is also a connection in the complex

vector bundle ðTM; �JJÞ (cf. Corollary 4.9)). Then the result follows from unique-

ness. For the first property (a), it suffices to show that if W = U + iV is a complex

vector field on M and �XX; �YY are holomorphic sections of TM, then W �HHð �XX; �YYÞ ¼�HHð �r W*

�XX; �YYÞ þ �HHð �XX; �rW�YYÞ; where *W is the complex conjugate of W and U, V 2

G(TM). Since (M, g, �, J) is Ka€hler, from Corollary 4.9 and Theorem 4.6,

W �HHð �XX; �YYÞ ¼ U�ggð �XX; �YYÞ þ iV�ggð �XX; �YYÞ þ iU ��ð �XX; �YYÞ � V ��ð �XX; �YYÞ¼ �ggð �rU

�XX; �YYÞ þ �ggð �XX; �rU�YYÞ þ i�ggð �rV

�XX; �YYÞ þ i�ggð �XX; �rV�YYÞ þ

þ i��ð �rU�XX; �YYÞ þ i��ð �XX; �rU

�YYÞ � ��ð �rV�XX; �YYÞ � ��ð �XX; �rV

�YYÞ:

But from Corollary 6.6,

�ggð �rV�XX; �YYÞ ¼ ��ð �rV

�XX; �JJ �YYÞ¼ ��ð�JJð �rV

�XXÞ; �YYÞ¼ ��ð �r�JV

�XX; �YYÞ¼ ��ð �r�iV

�XX; �YYÞ;

since �JJ �VV ¼ i �VV in the complex vector bundle ðTM; �JJÞ: Similarly,

�ggð �XX; �rV�YYÞ ¼ ��ð �XX; �riV

�YYÞ;�ggð �r�iV

�XX; �YYÞ ¼ ��ð �rV�XX; �YYÞ;

�ggð �XX; �riV�YYÞ ¼ ��ð �XX; �rV

�YYÞ:

Thus,

W �HHð �XX; �YYÞ ¼ �HHð �rU�XX; �YYÞ þ �HHð �XX; �rU

�YYÞ þ �HHð �r�iV�XX; �YYÞ þ �HHð �XX; �riV

�YYÞ¼ �HHð �r W*

�XX; �YYÞ þ �HHð �XX; �rW�YYÞ:

For the second property (b), let V 2 G(TM). Then, since �rV�XX is a section of the

complex vector bundle ðTM; �JJÞ for every holomorphic �XX 2 �ðTMÞ, from

Corollary 6.6,

�rVþiJV�XX ¼ �rV

�XX þ i �rJV�XX

¼ �rV�XX þ i�JJð �rV

�XXÞ¼ �rV

�XX � �rV�XX

¼ 0:

Thus, �rZ�XX ¼ 0 for every Z 2 G(TM)(0,1).


Conversely assume that the Koszul connection �r in ðTM; �ggÞ is a Hermitian

connection in ðTM; �JJÞ when extended complex linearly in the first argument. It

suffices to show that ð �rV��Þð �XX; �YYÞ ¼ 0 for every �XX; �YY 2 �ðTMÞ and V 2 G(TM).

Let V ¼ V þ i0 2 �ðTMÞC: Then V �HHð �XX; �YYÞ ¼ �HHð �rV�XX; �YYÞ þ �HHð �XX; �rV

�YYÞ from

property (a). Hence, V�ggð �XX; �YYÞ þ iV ��ð �XX; �YYÞ ¼ �ggð �rV�XX; �YYÞ þ �ggð �XX; �rV

�YYÞ þ i��ð �rV�XX; �YYÞ þ i��ð �XX; �rV

�YYÞ. Thus, it follows that ð �rV��Þð �XX; �YYÞ ¼ 0: Ì

THEOREM 6.8. Let g, � and J be compatible differential geometric structures ona complex manifold M, where (M, g) is a singular semi-Riemannian manifold andJ is the canonical almost complex structure of M. Let �r be the Koszul connectionin ðTM; �ggÞ. Then (M, g, �, J) is a pre-Ka€hler manifold and the following areequivalent:

(a) (M, g, �, J) is a Ka€hler manifold,(b) �r�JJ ¼ 0,

(c) �r�� ¼ 0,

(d) �dd �� ¼ 0.

Proof. To show that (M, g, �, J) is a pre-Ka€hler manifold, it suffices to show

that (M, �) is a singular almost symplectic manifold. For this, from Corollary

3.10, it suffices to show that �LLu�JJ ¼ 0 for every u 2 M?. Recall that �LLu is

tensorial in u 2 M? (see Definition 3.6) and M? is a holomorphic subbundle of

(TM, J). Thus, �LLu�JJ ¼ 0 for every u 2 M? since �LLU

�JJ ¼ 0 for every holomorphic

U 2 �ðM?Þ. Hence (M, g, �, J) is a pre-Ka€hler manifold. The equivalence of (a),

(b), (c) and (d) follows from Theorem 4.6 since M is a complex manifold. Ì

7. Degenerate Submanifolds of Nondegenerate Ka€hler Manifolds

In this section, we shall study the degenerate Ka€hler submanifolds of Ka€hler

manifolds of type (0, m U 1, n U 1). We shall call a Ka€hler manifold (M, g, �, J)

of type (0, m U 1, n U 1) a nondegenerate semi-Ka€hler manifold.

DEFINITION 7.1. Let (M, g, �, J) be a nondegenerate semi-Ka€hler manifold. A

complex submanifold S of M is called a degenerate Hermitian submanifold of

type (k0, m0, n0) if HjS is a Hermitian metric of type (k0 U 1, m0, n0) on S, where

H = g + i� is the Ka€hler metric of (M, g, �, J).

Note that if S is a degenerate Hermitian submanifold of a nondegenerate semi-

Ka€hler manifold, then since TS is invariant under J; g j S, � j S and J j S are

compatible differential geometric structures on S. Let S? be the real vector

bundle underlying the complex null space of H | S in TS (cf. Definition 4.3).


PROPOSITION 7.2. Let S be a degenerate Hermitian submanifold of type (k0,m0, n0) in a nondegenerate semi-Ka€hler manifold (M, g, �, J). Then (S, g j S, � j S,

J j S) is a Ka€hler manifold.

Proof. Since d� = 0, d� j S = 0 and, hence, (S, � j S) is a symplectic manifold

with nullity k0. Thus, it suffices to show that (S, g j S) is a singular semi-

Riemannian manifold (cf. Theorem 6.8). But, since S is a complex manifold,�LLu

�JJ ¼ 0 for every u 2 S?, where �JJ is the induced almost complex structure in�TSTS ¼ TS=S? (see the proof of Theorem 6.8). Hence it follows from Corollary

3.10 that (M, g) is a singular semi-Riemannian manifold. Ì

PROPOSITION 7.3. Every degenerate Hermitian hypersurface S of a nonde-generate semi-Ka€hler manifold (M, g, �, J) is totally geodesic.

Proof. Let l be the Levi-Civita (Koszul) connection of (M, g) and let ±(TS)

be the orthogonal bundle to TS in TM with respect to g. Then, since the (real)

ranks of ±(TS) and S? are equal to 2, ? ðTSÞ ¼ S? � TS. Hence ? ðS?Þ ¼ TS,

where ? ðS?Þ is the orthogonal bundle to S? in TM with respect to g. Thus it

suffices to show that g(lX Y, U) = 0 for every X, Y 2 G(TS), U 2 �ðS?Þ to show

lX Y 2 *(TS) for every X, Y 2 G(TS). Indeed, since LUg ¼ 0 (cf. Proposition 7.2),

gðrXY;UÞ ¼ XgðY;UÞ � gðY;rXUÞ¼ �gðY;rUX þ ½X;U�Þ¼ �gðY;rUXÞ � gðY; ½X;U�Þ¼ �UgðY;XÞ þ gðrUY;XÞ � gðY; ½X;U�Þ¼ �UgðY;XÞ þ gðrYU þ ½U; Y�;XÞ � gðY; ½X;U�Þ¼ �UgðY;XÞ þ gðrYU;XÞ þ gð½U; Y�;XÞ � gðY; ½X;U�Þ¼ �ðLUgÞðX;YÞ þ gðrYU;XÞ¼ gðrYU;XÞ¼ YgðU;XÞ � gðU;rYXÞ¼ �gðU;rXY þ ½Y;X�Þ¼ �gðU;rXYÞ:

Hence, 2g(U,lXY) = 0. Ì

DEFINITION 7.4. Let (M, g, �, J) be a nondegenerate semi-Ka€hler manifold of

dimension 2n. An n-dimensional submanifold S of M is called a Lagrangian

submanifold of (M, g) if g | S = 0.

Remark 7.5. Let (M, g, �, J) be a nondegenerate semi-Ka€hler manifold and Sbe a Lagrangian submanifold of (M, g, �, J). Then, since the dimension of S is the

half of the real dimension of M, (M, g, �, J) is necessarily of type (0, m, m) and

the dimension of S is equal to m.


PROPOSITION 7.6. Let (M, g, �, J) be a nondegenerate semi-Ka€hler manifold.Then every Lagrangian submanifold of (M, g, �, J) is a totally geodesic complexsubmanifold of (M, g, �, J).

Proof. First we show that S is a complex submanifold. Since the dimension of

S is half of the real dimension of M and g | S = 0, ±(TS) = TS = S±, where ±(TS)

is the orthogonal bundle to TS with respect to g. Thus, if X 2 G(TS) then, JX 2G(TS) iff g(JX, Y) = 0 for every Y 2 G(TS). But, since 0 = �(X, Y) = g(JX, Y) for

every X, Y 2 G(TS), JX 2 G(TS) for every X 2 G(TS). Thus, S is a complex

submanifold of M. To show that lXY 2 G(TS) for every X, Y 2 G(TS), it suffices

to show that g(lXY, Z) = 0 for every X, Y, Z 2 G(TS). Indeed, since LUg ¼ 0 for

every U 2 �ðS?Þ ¼ �ðTSÞ ¼ �ð? ðTSÞÞ, it can be shown as in the proof of

Proposition 7.3 that 2g(lXY, Z) = 0. Ì

References

1. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. 2, Interscience,

New York, 1969.

2. Kupeli, D. N.: Degenerate manifolds, Geom. Dedicata 23(3) (1987), 259Y290.

3. Nomizu, K.: Remarks on sectional curvature of an indefinite metric, Proc. Amer. Math. Soc.89(3) (1983), 473Y476.

4. Poor, W. A.: Differential Geometric Structures, McGraw-Hill, New York, 1981.


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