ON THE SEMICLASSICAL DENSITY OF STATES OF QUASI-INTEGRABLE MECHANICAL SYSTEMS

STEFANO ISOLA, 1 ROBERTO LIVI 2'3 and STEFANO M A R M I 4

1 Universitd di Bologna, Dipartimento di Matematica, 40127 Bologna, Italy. a Universit6 di Firenze, Dipartimento di Fisica, 50125 Firenze, Italy.

3INFN, Sezione di Firenze. 4Universit6 di Firenze, Dipartimento di Matematica 'U. Dini', 50100 Firenze, Italy.

(Received: 20 March 1992; accepted in revised form: 17 November 1992)

ABSTRACT. For a class of quasi-integrable mechanical systems, we show that in the semiclassical limit the Bohr- Sommerfeld quantization applied to successive truncations of the Birkhoff series and a power series expansion of Weyl's formula give the same asymptotic number of states below a given energy.

SOMMARIO. Si mostra che per una classe di sistemi quasi integrabili, nel limite semiclassico, la quantizzazione di Bohr-Sommerfeld applicata a successivi troncamenti della serie di Birkhoff e l'espansione in serie di potenze della formula di Weyl danno lo stesso andamento asintotico per la densita' degji stati integrata.

KEY WORDS. Quasi-integrability, Semiclassical limit, Theoretical mechanics.

1. I N T R O D U C T I O N

The quantum mechanics for a dynamical system in a compact space yields a set of discrete energy levels

E1 <~ E2 ~ ""El <~ "'"

which contains some characteristic information on the geometrical and dynamical features of the classical system. From this set one can compute the spectral staircase

N(E) -- ~ O(E--Ei) (1.1) i

whose derivative gives the density of states

d d(E) = ~-E N(E). (1.2)

On the other hand, the asymptotic behaviour of this function can be directly related to the volume in classical phase space, according to a famous theorem by Weyl which goes back to the very beginning of quantum theory [1].

Let us see how it works in a simple case. One of the most frequently used model systems is that of a particle moving freely in a finite domain f2, with boundary 8f~. The shape of the boundary then determines the nature of the classical motion whereas the quantum spectrum is given by the eigenvalue problem

A(0+k2q~=0 inf~ (1.3)

~o = 0 on 8~

where A is the Laplacian in ~z and k = ( 2 x / ~ / h . In this case Weyl's formula writes

d(f~)k z N(E) - 4 ~ + n(E) (1.4)

where ~4(fl) is the area of f~ and n(E) = (E).

Meccanica 28:309 314, 1993 © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

Further corrections due to the boundary may be in- cluded if, instead of N(E), one considers an averaged counting function 57(E) (see, e.g., [2]).

Weyl's formula (1.4) has a remarkable interpretation in terms of classical mechanics. Since the potential energy V is zero inside the domain fL the kinetic energy of the particle is constant and the momentum is given by ]p[e= 2mE. Then the total volume of phase space with energy less than E is given by Y.(E)= 2rcmE~C(fl) so that (1.4) becomes N(E) ~- E(E)fl2nh) 2 for E ~ oo.

An immediate generalization is that of a quantum particle moving freely on a two-dimensional Riemannian manifold (YL g). The Schr6dinger equation is then written as

A~0 + k2~o = 0 (1.5)

where

1 8 i ( dx/dx/dx/dx/dx/dx/d~ gljSjq~) (1.6) A0q~ -

is the Laplace-Beltrami operator corresponding to the Riemannian metric g. In this case, Weyl's formula (1.4) holds with d(f~) replaced by

~4(f~, g) = 5a dr,(q) (1.7)

where dv o denotes the canonical Riemannian measure. Let us now see that analogous expressions hold for a

class of mechanical systems for which the motion inside a domain f~, instead of being free, is governed by a potential function V.

Consider a Hamiltonian describing a mechanical system with d freedoms

H(p, q) = 7 + V(q) (1.8) i=1

310 S T E F A N O I S O L A E T AL.

where V is a measurable function on ~a. Let D(E) = Z(E)/(2r~h) a where

E(E)= f dp d q = f O(E-H(p, q))dp dq (1.9) Jn (p,q)~<E 3

is the phase space volume enclosed by the constant energy manifold. We then have the following ([3, p. 275])

T H E O R E M 1. Let V be a measurable function on ~d (d >12) obeyin9

cl(Iql N- 1) ~< V(q) <~ C2(Iq]N + 1) (1.10)

IV(q) -- V(q')l ~< c3[max{lql, Iq' I}] N- llq - q']

for some N > i and suitable constants q, c2, c3 > O. Then

N(E) lim - 1. (1.11) ~ D(E)

In this paper we deal with a class of potentials which are polynomials of even degree N ~> 2, so that they obviously satisfy the hypotheses of the theorem.

A second remark is {q ~ Re I V(q) <~ E}, then one following way:

Z(E)=fadqf:<.2tE_v(q)l dp

the following: let fl = can compute Z(E) in the

- d/2r(d/2) [ 2 ( E - V(q))] .2 dq

=(2E,a/2CafaI1-V(q)la/2d q (1.12)

=fp2<,2edPfn dvo(q)

where Ca is the volume of the d-dimensional unit sphere

and

dvg = I 1 - V(Eq) ]d/2 dql...dqa (1.13)

is the Riemannian measure corresponding to the Jacobi

metric

. [, .,4,

It is well known that the metric (1.13) naturally appears when the Hamiltonian flow is interpreted, through the Maupertuis principle and a suitable reparametrization of time, as a geodesic flow on the Riemannian manifold (fl, 9) [4]. Our hypotheses on the potential imply that such a manifold is compact with (piecewise) smooth boundary af~ = V- X(E) c ~a, for any E > infq V(q). Moreover, it is not difficult to realize that its Gaussian curvature is given by the expression

(1 -- v)Av + (Vv) 2 K = (1.15)

2(1 - v ) 3

where v(q)= V(q)/E. Then, the statement (1.11) can be interpreted as a Weyl

formula for the eigenvalue distribution of a d-dimensional spectral problem like (1.5), (1.6), where 9 is the Jacobi metric defined in (1.14).

Furthermore, as we shall see in a moment, if one deals with a completely integrable Hamiltonian system, the asymptotic behaviour of N(E) can also be obtained by passing to action and angle variables and computing the volume enclosed by the energy surface in the positive action space (see, e.g., [5], [6], [7]) so that Weyl's formula becomes essentially equivalent to the 'old quantum theory' prescription.

The aim of this paper is to show that this corre- spondence can be extended to the case of quasi-integrable Hamiltonian systems, and this is achieved by comparing a power series expansion of the phase space volume in terms of the perturbation parameter to the result obtained by successive truncation of the Birkhoff series for the Hamil- tonian under consideration.

Section 2 is devoted to a brief discussion of the completely integrable case in order to properly introduce the main result, which is presented in Section 3. In Section 4 we give an example where the calculations can be fully performed, as a clarifying reference. Section 5 contains some further remarks and the conclusions.

2. I N T E G R A B L E SYSTEMS

In this case it is possible to find a canonical transformation to action and angle variables (I, q~) = <g(p, q), such that the new Hamiltonian does not depend on the angles:

H(~- 1(1, 4~)) = H(I). (2.1)

In these variables the phase space appears as the direct product ~ x T a, where ~ ~ R a is an open set and T a is the d-dimensional torus. It is therefore easy to obtain an alternative expression for the phase space volume, as

follows:

E(E) = f O(E- H(p, q)) dp dq

= f O(E--H(I)) dI dO

(2~) a f dI 3/t (i) < E

- (2~)nJ(E). (2.2)

The function J (E) represents the volume contained by the hypersurface H(I)=E in ~a+. It then also gives the as- ymptotic value (i.e. when E ~ ~ ) of the number of Z~- lattice points inside this hypersurface. Hence, the identity

Y(E) (2.3) D(E)- ha

SEMICLASSICAL DENSITY OF STATES 311

is an account of the statement that Weyl's formula for the spectral problem described in Section 1 and the Bohr - Sommerfeld quantization condition

Ek = H(I = kh) (2.4)

give the same asymptotic result for the function N(E).

3. QUASI-INTEGRABLE SYSTEMS

We suppose now that a perturbation parameter e can be introduced into the Hamiltonian in such a way that for e = 0 it degenerates into a completely integrable Hamil- tonian Ho, and also it may be expanded in a series

H = Ho + all1 + aZH2 + " " (3.1)

which converges for sufficiently large values of the coor- dinates and momenta. In other words, we are considering those situations where the canonical theory of perturba- tions is naturally adopted to obtain approximate de- scriptions of the dynamics [8].

More precisely, we consider a Hamiltonian system of the form

2 2 2 H(p, q, e) = PJ + ooj qj

j=~ 2

where (p,q)eN2 d, e is a

polynomial of degree

+ ef(q) (3.2)

small parameter and f is a N, NI>3 . Set V(q)=

]Fad 2 2 j = ~ coj qj/2 + ef(q). We assume that:

(H1) V(0) = 0; (H2) limq~ + ~ V(q) = + 0o for all i = 1 . . . . . d;

c~2f (H3) V f(0) = 0, ~ (0) = 0, for all i, j = 1 . . . . . d.

(H1) is simply a normalization condition, whereas (H2) ensures that the hypotesis of Theorem 1 is satisfied, so that, in particular, the constant energy manifolds

m~,~ = {(p, q)e ~2"lH(p, q, e)= E} (3.3)

are compact for any e. Let E > 0 be fixed. The phase space volume within M~.~ is defined by

E(E, e)= f dp d q = ~ O(E-H(p , q, a))dp dq. (3.4) dn (p,q,e) ~< E d

Since ME. ~ depends analytically on e as it varies in a neighborhood of the origin, the function e F-. I£(E, e) is analytic at e = 0 and can be developed into a convergent power series

E(E, e ) = ~ ak(E)e k. (3.5) k = O

Clearly, ao(E)=(2E)dC2a/col .. <%. The convergence radius e~(E) of the series development is implicitly fixed by the critical values of H: at those values the constant energy manifold may not depend analytically on e and its to- pology may change. Note that lee(E)] ~ + ov as E ~ 0.

For e = 0, the Hamiltonian (3.2) describes d independent harmonic oscillators and is obviously integrable. Action- angle variables (1, ~b) s Re+ x T d can be introduced:

ql = ~ sin ~b i (3.6)

pi = ~ cos ~i

and (3.2) becomes

d

H(I, O, e) = ~ cOklk + eF(I, (O) (3.7) k = l

where F(I, e~)= f(p(I , ~), q(1, q~)) is a trigonometric poly- nomial of degree N. Canonical perturbation theory can now be applied to our system: one looks for a family cg, of analytic canonical maps, which depends analytically on e, for e small enough, defined on an open subset 2 x T d and such that cg~ iden t i t y as e ~ 0 . If (J, O) = Cg~(I, c)), and given a positive integer r >~ 2, one requires that

(HoCg~-a)(J, O, e) = W(°(J, O, e)+ Rr(J, O, e) (3.8)

where W (') has a prescribed dependence on the angles 0, according to the resonant or non-resonant nature of the frequencies co of the unperturbed motion. Since the perturbation F has only a finite number of harmonics, at all finite orders r of perturbation theory one deals with trigonometric polynomials of degree at most rN. We assume that there exists a positive constant ~, and a codimension d - 1 module J / of Z d, l = 0 . . . . . d - 1 such that

]co.v[ ~> a~ for all vCJ/g, Iv I <<. rN, (3.9)

where • denotes the usual scalar product in Nd and

Ivl = E]= 1 Ivj]. One then requires that W(')(J, O, e) depends on the angles only through resonant harmonics, i.e.

W(')(J, 0, e) = co- J + ~ ~Wk(J, 0) k=l (3.10)

Wk(J, O)= Z lfVk.v(J) eg~'°" v 6 ~ /

Let us now come to a detailed treatment of such transformation, with the necessary convergence estimates. To this end we will closely follow [9] and we need to introduce some notations and the norms used.

As usual in canonical perturbation theory with analytic- ity assumptions, in order to use Cauchy's inequality for the derivatives, one complexities the domain of the action variables. Let p > 0, then

~o = { IeCd; ]Ik--Ik[ ~< p, I ' e ~ , k = 1 . . . . . a} (3.11)

denotes the union of polydisks of radius p centered on any point of N. Given 0 < 6 < p, we denote by N o - 6 the set of points of ~o whose distance from the boundary is at least 6. We define @p = ~o x T d and 9 o - 6 = ( ~ o - J )x T d. We consider real analytic functions g on 9 o

g(1, ~ ) = 2 (t~(I) ei~'4' (3.12) v e Z a

312 STEFANO ISOLA ET AL.

and the norms

II~II~ = Y~ t0,,1~ v ~ d

10~l.~. -- sup 10~(I,I. Ie~/¢

Then, one has [9] the following

(3.13)

T H E O R E M 2. For any 6<p/2 and r>~2 there exists a canonical transformation from ~ - 2 6 to No, which brings the Hamiltonian (3.7) in the normal form (3.8). The trans- formed Hamiltonian converges for e < e,*, where

36o~ e,* = lONr 3 [IFII~, (3.14)

and for the remainder R~ one has the estimate

(/~ ~ r+ l 'lRr 'le,-Z, ~< (1 - - ~@) - ~ ~e~-'*j IIHI,~o. (3.15,

Some details on the effective construction of the normal form (3.8) will be reported in the next section, along with an explicit example.

We now introduce the functions

f w dd dO~...dO~ ~('~(E, 13) -- ~'~(J,0,,)~<e (3.16)

where (01,. . . , 0~) are the resonant angles. For each r >~ 0, these functions are defined and analytic in 13 provided e < 13,*. Clearly, (2~)a-~¢(~)(E,13) is the volume within the manifold W~')(J, O, 13) = E. The Hamiltonian W(~)(J, O, e), obtained by truncating the Birkhoff normal form at order r, approximates (3.2) at order r. In particular, if there is no resonance relation among the unperturbed frequencies co, i.e. if ~ / = {0}, this gives an integrable system approximat- ing (3.2) on a domain which shrinks as r ~ oe. One has the following:

PROPOSITION. The formal series

(2~)a-' ( J{°)(E' e )+ ,=o ~ ~r~'+*)(E' e)-J{')(E' ~)) (3.17)

has a power series expansion w.r.t. 13 which termwise with (3.5).

Proof One obviously has the identity

coincides

E(E, ~) = (2rt) ~ - ~¢~'~(E, e) +

+(fw,,,(j.o,,)+~,(j,o,~)<~ dJ d 0 - (2r0a-~"~(E, e))"

Then, since our norm gives an upper bound to the maximum norm, the term between parentheses is es- timated as 21[R,[l~,_2~.volume(M~.~. ). The estimate pro- vided in (3.15) concludes the argument.

Furthermore, observe that under the additional assumption that f is a homogeneous polynomial (of

degree N), the Hamiltonian (3.2) obeys the scaling law

(q~, pl) ~ (eqi, ~p~), i = 1 . . . . , d

E ~ eaE (3.18)

t3 ---~ 13/0~ N - 2.

Hence, the classical dynamics depends only on the parameter

fl = E~ ~/~'- ~. (3.19)

It is then easy to realize that in terms of this parameter the convergence of (3.5) is assured as long as 1/~1 < 1, whereas the transformed Hamiltonian (3.8) converges for [ill < fl'*, where fl'* is given by inserting (3.14) in (3.19) (notice that fl,* 4 0 as r --* ~). For a more general polynomial function f (not necessarily homogeneous) the same result holds true in the limit E ~ ~ . However, according to the discussion given in Section 1, this is the limit to be taken in order to compare the phase space volume with the number of quantum levels below a given energy (once h is kept fixed). In particular, if ~/¢' = {0} then W~(J, 0) = W~(J) (see (3.10)). Hence, if one introduces the function ~C(*I(E,e) which gives the number of Z~+-lattice points contained inside the surface Wt')(J, 13)= E, then

~C(°(E, 13) lim = 1 (3.20) ~-, ~o ~¢('~(E, ~)

provided fl sufficiently small so that everything is well defined. Finally, putting together the above results and Theorem 1 we have the

COROLLARY. Assume that ~//d= {0} and till < fl'*. Let ~V'~r)(E, 13) be as above. Then

~4/~r)(E' e)/ha ~ l+(9(f f +1) as E-+ oo (3.21) ~v(~

where N(E) is the staircase function (1.1) .['or the quantiza- tion of (3.2).

REMARK. Since ~¢/, = {0}, the function ~/g'(')(E, 13)/h ~ fur- nishes the number of eigenvalues below E obtained by applying Bohr-Sommerfeld quantization rule (2.4) to the truncated Birkhoff series W(')(J, e).

4. A N E X A M P L E

We first consider the simple case of two harmonic oscil- lators (d = 2) with equal frequency o), whose Hamiltonian is

2 2 2 2 _ ~oz q l + qz. /4"(p, q) Pl +Pz + (4.1)

2 2

In action and angle variables, given by (3.6), it writes H=o2(I1+12), whose quantization gives Et=h~o(l+l), where l = m + n, m, n ~ Z + w {0}, so that the function N(E)

SEMICLASSICAL DENSITY OF STATES 313

can be easily computed:

[e/hoe] ([ E/h0)] + 1)([ E/h0)] + 2) N(E)= E ( / + 1 ) - -~

t=o 2

E 2 3E 2h20)~ + ~ as E --* oo (4.2)

where the first term in the r.h.s, is just the area enclosed by the line 0)(11 + I 2 ) = E in ~2+ divided by h 2.

On the other hand, the 'membrane' (f~, g) is the disc of

radius r(E)=,¢/~-/0) endowed with the metric (in polar coordinates (r, ~)):

( 0)2r2~ dg 2 = 1 - - - ~ ] (dr 2 q- r 2 d~ 2) (4.3)

According to (1.15), its Gaussian curvature is

0)2 /E 0) 2

K - (1-0)2r2/2E) ~> - E > 0 (4.4)

so that K ~ oo approaching the boundary 8D. Finally, an easy computation gives the total area ~¢(~, g )= 7rE/0) 2. Inserting this expression in Weyl's formula (1.4) one recovers the leading term of (4.2).

Let us now consider the Hamiltonian

H(p, q, ~) = Ho + eq~q 2 (4.5)

with Ho being a non-degenerate version of (4.1), i.e.

p2.4_ p2 q2 d- c02q 2 H°(P' q) = 2 -t 2 (4.6)

and 0)e EkQ. Again using (1.15), it is easily seen that the Gaussian curvature of (f~, 9,) is positive for any value of e.

Let us first compute ~¢(f~,g~) for the system (4.5).

Introducing the coordinates ql = x ~ r cos 6, q2 =

(V/2--E-/0))r sin 6, we find that the equation of the boundary Of~ is

0)2 r2(6) - 2eE sin z 26 (x/1 + fl sin2 2~, - 1) (4.7)

where fi = 4~E/0) is the physical parameter which controls the dynamics of the system. Then,

sff(f~.,g~) 2 E f o ' ~ d O i ~ ( ° ' ( fir4 s 4 4 2 0 ) = - - 1--r 2 - - rdr. 0) do

(4.8)

This integral can be evaluated explicitly and one finds

32Ex/T-+fl((l+~)K( f l ~ ) - ~¢(f~' g~) - 90)fl

- ( l + 2 f l ) E ( f l ~ ) ) (4.9)

where K and E are elliptic integrals of the first kind. Equation (4.9) provides the volume accessible to the dynamics for any value of e. However, for e small enough one can expand the above expression around e = 0, as in

(3.5), obtaining

( ~2 3 ) d ( f ~ , g,) = d o 1 - + ~ f l 2 + C ( f i 3 ) (4.10)

where d o = roE~o) is the area corresponding to the unper- turbed Hamiltonian (4.6).

Hence, we find

E ( ~2 3 f12 q- (-0(f13)) (4.11) D(E) = ~ do(E ) 1 - + / y 8

This expansion converges for ]ill < 1, as one can find by direct computation, or by means of the following argu- ment. The critical points of the Hamiltonian (4.5) are the solutions of the equations p1=p2=0 , ql( l+2eq2)=O, q2(0)2 + 2eq 2) =0. Therefore the critical set consists of the point (0, 0, 0,0) and of the curves (0, 0, q~ = - co2/2~, q2=q~/0)2). The corresponding critical values for the energy are E = 0 and E=-0 )2 /4e . The function D(E) will depend analytically on the parameters as long as the critical values are not crossed, since only at those values the constant energy manifold may not depend analytically on E and may bifurcate. Note that the value Ifl[= 1 corresponds to the non-zero critical values of E.

We now compare (4.11) with the corresponding ex- pression obtained via canonical perturbation theory.

In terms of action-angle variables of (4.6), (4.5) becomes

4e H = I t +0)12 q- - - 1112 sin 2 ¢1 sin2 (~2, (4.12)

0)

then, in order to construct the Birkhoff normal form (3.8), we shall follow the strategy (see, e.g., [10]) of constructing a family of generating functions S(r)(dp, J, e) of canonical transformations cg~r) such that

8S (~) 8S (,) I i - 6~) i , 0 i -- t3ji , (4.13)

we put

S(~)(q~, J, e) = ~, ekSk((a, J) (4.14) k=0

where S O is the identical transformation ZidpiJ~, and $1, $2 . . . . are periodic in the 01's. The Hamiltonian (3.7) in the new variables is therefore

~s('(¢, J, ~) (~s(r'(¢, J, ~) ) 0)" 80 + eF \ -~¢ , dp (4.15)

where co here denotes the frequency vector. Since by hypothesis everything is analytic we can expand the above expression in powers of e and obtain a number of dif- ferential equations by imposing the coefficients of like powers of e to be ~b-independent. These equations have the form

0)" J = Wo(J ) ,

8Sk (4.16) 0 ) ' ~ = Wk(J) - N,(O, J)

314 STEFANO ISOLA ET AL.

where N k is a po lynomia l in the variables 0S~/~49, 1 = 1 . . . . .

k - 1. In part icular, since the Sk'S are periodic in the angles

~bi's, one finds

Wk(J ) = <Nk(C~, J)> (4.17)

where the angle brackets denote t ime average over the unper turbed motion. In our example, the generat ing func-

tion is found to be

S = ~ J~c~ i + eS1 + "'" (4.18) i

where

2 J1J2 2 sin 2~b 1 + - - sin 2~b 2

Sa(q~' J) - 40) 0)

1 1 sin 2(q~x + q~2)) + ~ sin 2(~bl-~b2) + ]-~0)

(4.19)

so that to the second order in e we obtain the normal form

8 W(2)( J, ~) = J1-4- °)J2 + - - J1J2

0)

82J1J2 ['J1 J2 J z - J 1 J l q-J2 (4.20)

092 ~ - ~ + ~ - - 4 8 (1-0) ) ~-8(1+0)~ J We can now compute the functions (3.16) and for k = 1, 2

we get

J ° ) ( E , e) = ~ 1 - , (4.21)

= 1 - + ] S g / 3 2 ,

where/3 = 4eEl0). As expected, they exactly coincide with the corresponding part ial sums in (4.11), when divided by h 2"

5. C O N C L U S I V E REMARKS

Some questions deserve further attention. First, for the class of potentials we have considered, quan tum correc-

tions to the Bohr -Sommer fe ld rule applied to the classical Birkhoff series have been explicitly constructed in [11] as convergent expansions in powers of h. It would be of great interest to unders tand the role of such corrections in

improving the asymptot ic est imate (3.21). Another quest ion which could be an interesting chal-

lenge for future work is the following: according to the

discussion given in Section 1, to the geodesic flow on the

manifold (f2, g), corresponding to the natural Hami l ton ian system (1.8), is associated the spectral p rob lem (1.5) in such a way that the resulting eigenvalues can be thought of as the eingenfrequencies of the ' m e m b r a n e ' (f~, g), whose geomet ry is determined by g and whose volume is fixed by

the classical energy E (and eventually by other classical parameters such as ~). On the other hand, we have seen that this spect rum and the energy spect rum obta ined by

the quant iza t ion of (1.8) are asymptot ica l ly related, in so far as they give the same leading behaviour of N(E) when

E approaches infinity. One then wonders abou t possibly closer relat ionships between the two. This would be

desirable also for the spectral geomet ry of (f~, g) yields often interesting connect ions with some geometrical and

dynamical features of the classical system (see, e.g., [12]).

A C K N O W L E D G E M E N T S

The authors wish to thank S. Graffi, H. Kantz , Ya. G. Sinai and M. Vittot for interesting discussions and useful com-

ments. This work has been part ial ly done at the Inst i tute for Scientific In terchange (I.S.I.), Villa Gual ino, Torino, which is warmly acknowledged for its financial suppor t and kind hospitality.

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