boris.unibe.ch · proceedings of the edinburgh mathematical society (2008) 51, 711–750 c...

Proceedings of the Edinburgh Mathematical Society (2008) 51 711ndash750 ccopyDOI101017S0013091506000150 Printed in the United Kingdom

SPECTRAL THEORY OF THE KLEINndashGORDONEQUATION IN KREIN SPACES

HEINZ LANGER1 BRANKO NAJMAN2lowast AND CHRISTIANE TRETTER3

1Institut fur Analysis und Scientific Computing Technische Universitat WienWiedner Hauptstrasse 8ndash10 1040 Wien Austria (hlangeremailtuwienacat)

2Department of Mathematics University of Zagreb Bijenicka 30 41000 Zagreb Croatia3Mathematisches Institut Universitat Bern Sidlerstrasse 5

3012 Bern Switzerland (trettermathunibech)

(Received 19 January 2006)

Dedicated to the memory of Peter Jonas

Abstract In this paper the spectral properties of the abstract KleinndashGordon equation are studied Themain tool is an indefinite inner product known as the charge inner product Under certain assumptions onthe potential V two operators are associated with the KleinndashGordon equation and studied in Krein spacesgenerated by the charge inner product It is shown that the operators are self-adjoint and definitizable inthese Krein spaces As a consequence they possess spectral functions with singularities their essentialspectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvaluesof finite algebraic multiplicity which are symmetric to the real axis One of these operators generatesa strongly continuous group of unitary operators in the Krein space the other one gives rise to twobounded semi-groups Finally the results are applied to the KleinndashGordon equation in Rn

Keywords KleinndashGordon equation Krein space charge inner productself-adjoint operators in Krein spaces definitizable operators

2000 Mathematics subject classification Primary 81Q05 47B50Secondary 47D03 35Q40

1 Introduction

The KleinndashGordon equation((part

parttminus ieq

)2

minus ∆ + m2)

ψ = 0 (11)

describes the motion of a relativistic spinless particle of mass m and charge e in anelectrostatic field with potential q (the velocity of light being normalized to 1) here ψ isa complex-valued function of x isin R

n and of t isin RIf in (11) we replace the uniformly positive self-adjoint operator generated by the

differential expression minus∆ + m2 in the Hilbert space L2(Rn) by a uniformly positivelowast Sadly Professor Branko Najman died in August 1996

711

httpswwwcambridgeorgcoreterms httpsdoiorg101017S0013091506000150Downloaded from httpswwwcambridgeorgcore University of Basel Library on 30 May 2017 at 135051 subject to the Cambridge Core terms of use available at

source httpsdoiorg107892boris36873 | downloaded 1112020

712 H Langer B Najman and C Tretter

self-adjoint operator H0 in a Hilbert space H and the operator of multiplication withthe function eq by a symmetric operator V in H we obtain the abstract KleinndashGordonequation ((

ddt

minus iV)2

+ H0

)u = 0 (12)

where u is a function of t isin R with values in H Equation (12) can be transformed into afirst-order differential equation for a vector function x in an appropriate space formallygiven by H oplus H and a linear operator A therein

dx

dt= iAx (13)

However this is in general not possible with a self-adjoint operator A in a Hilbert spaceIn the literature two inner products have been associated with the KleinndashGordon equa-

tion (11) one of them representing the charge and the other one representing the energyof the particle The energy inner product is used in numerous papers to study the spec-tral and scattering properties of (11) see for example [511121620282935ndash3741424950] The charge inner product has been suggested by the physics literature(see [13]) it was first used in the pioneering work of Veselic [45ndash47] and subsequentlyin the papers [343848] the unpublished manuscriptlowast [26] as well as in [917ndash19] Itis also called the number norm since it is related to the number operator in the theoryof second quantization (see [3])

The reason for the preference of the energy inner product 〈middot middot〉 may be that it is positivedefinite and generates a Hilbert space if the potential V is small with respect to H

120

this can be seen from its formal definition

〈xxprime〉 =

((H0 minus V 2 0

0 I

)xxprime

)= ((H0 minus V 2)x xprime) + (y yprime) (14)

for suitable elements x = (x y)T xprime = (xprime yprime)T of H oplus H where (middot middot) denotes the scalarproduct in H The charge inner product [middot middot] however is always indefinite it is definedon elements x = (x y)T xprime = (xprime yprime)T of H oplus H by a relation of the form

[xxprime] =

((0 I

I 0

)xxprime

)= (x yprime) + (y xprime) (15)

Therefore it is negative on an infinite-dimensional subspace (if H is infinite dimensional)and hence leads to a so-called Krein space At first glance these indefinite structures seemto be less convenient from the mathematical point of view However they allow deeperinsight into the spectral properties of the KleinndashGordon equation eg by providing aclassification of the points of the spectrum into points of positive negative or neutraltype and sufficient conditions for the existence of corresponding strongly continuousgroups of operators which are unitary with respect to the indefinite inner product (15)

lowast This manuscript dating back to the late 1980s was the starting point for the present paper andalso for the papers [1827]


KleinndashGordon equation in Krein spaces 713

In this paper we associate two operators A1 and A2 with the abstract KleinndashGordonequation (12) Formally both operators arise from the second-order differential equa-tion (12) by means of the substitution

x = u y =(

minus iddt

minus V

)u (16)

which leads to a first-order differential equation (13) of the form

dx

dt= i

(V I

H0 V

)x (17)

for the vector function x = (x y)T However we consider the block operator matrixin (17) in two different Krein spaces induced by the charge inner product (15) andwe prove the self-adjointness of the corresponding operators A1 and A2 in these Kreinspaces

The main results of this paper concern the structure and classification of the spectrumfor A1 and A2 the existence of a spectral function with singularities the generation of astrongly continuous group of unitary operators the existence of solutions of the Cauchyproblem for the abstract KleinndashGordon equation (12) and an application of these resultsto the KleinndashGordon equation (11) in R

n The main tools for this are techniques fromthe theory of block operator matrices and results from the theory of self-adjoint operatorsin Krein spaces

The paper is organized as follows in sect 2 we present some basic notation and definitionsfrom spectral theory and we give a brief review of results from the theory of self-adjointand definitizable operators in Krein spaces In sect 3 we introduce the first operator A1associated with (17) which acts in the Hilbert space G1 = H oplus H Equipped with thecharge inner product (15) the space G1 becomes a Krein space which we denote by K1The block operator matrix in (17) is formally symmetric with respect to the charge innerproduct [middot middot] since for x isin (D(V ) cap D(H0)) oplus D(V ) xprime isin H oplus H[(

V I

H0 V

)xxprime

]=

((H0 V

V I

)xxprime

) (18)

We show that if V is relatively bounded with respect to H120 then the block operator

matrix in (17) is essentially self-adjoint in K1 and we denote its self-adjoint closureby A1 Using a certain factorization of A1 minus λ λ isin C we relate the spectral propertiesof A1 to those of the quadratic operator polynomial L1 in H given by

L1(λ) = I minus (S minus λHminus120 )(Slowast minus λH

minus120 ) λ isin C

where S is the bounded operator S = V Hminus120

In sect 4 we define the second operator A2 associated with (17) in the more complicatedHilbert space G2 = H14 oplus Hminus14 here Hα minus1 α 1 is a scale of Hilbert spacesinduced by the fractional powers Hα

0 of the uniformly positive operator H0 We equip G2



with the charge inner product (15) where now the brackets on the right-hand sideof (15) denote the duality

(x y) = (Hα0 x Hminusα

0 y) x isin Hα y isin Hminusα

between the spaces Hα and Hminusα for α = 14 and α = minus 1

4 the corresponding space K2 is aKrein space An analogue of formula (18) in H14 oplusHminus14 shows that the block operatormatrix in (17) now considered as an operator in G2 is formally symmetric with respectto the charge inner product [middot middot] If the resolvent set ρ(L1) is non-empty we associate aself-adjoint operator A2 in the Krein space K2 with the block operator matrix in (17)

The operator A2 was first considered by Veselic [45] (see also [344647]) He showedthat under his assumptions A2 is similar to a self-adjoint operator in a Hilbert space Theoperators A1 and A2 were also introduced by Jonas [18] using so-called range restriction(see [17]) In [18] and in [45] it was supposed that V H

minus120 is compact The operator A2

was also studied in [934] for the cases when either V Hminus120 is compact or V H

minus120 lt 1

The main results of the present paper are proved under the more general condition

S = V Hminus120 = S0 + S1 S0 lt 1 S1 compact (19)

Under this assumption in sect 5 we study and compare the spectral properties of theoperators A1 and A2 We show that A1 and A2 are definitizable (for the definition ofdefinitizability see sect 2) that their spectra essential spectra and point spectra coincideand are symmetric to the real axis that their essential spectra are real and have agap around 0 and that the non-real spectrum consists of a finite number of complexconjugate pairs of eigenvalues of finite algebraic multiplicity this number is bounded bythe number κ of negative eigenvalues of the operator I minus SlowastS in H As a consequenceof the definitizability the operators A1 and A2 possess spectral functions with at mostfinitely many singularities

At the end of sect 5 we compare the results for A1 with results for another operatorassociated with the KleinndashGordon equation in [27] This operator A arises from thesecond-order differential equation (12) by means of the substitution

x = u y = minusidu

dt (110)

which leads to a first-order differential equation of the form

dx

dt= i

(0 I

H0 minus V 2 2V

)x (111)

The operator A formally given by the block operator matrix in (111) acts in the spaceG = H12oplusH it is defined if V is H

120 -bounded and 1 isin ρ(SlowastS) These two assumptions

guarantee that a self-adjoint operator H = H120 (I minus SlowastS)H12

0 can be associated withthe entry H0 minus V 2 in (111) Under the additional assumption 19 the space K is aPontryagin space and A is a self-adjoint (and hence definitizable) operator in K We



prove that the spectra essential spectra and point spectra of A and of A1 (and hencealso of A2) coincide

In sect 6 we show that A2 generates a strongly continuous group (eitA2)tisinR of unitaryoperators in the Krein space K2 Hence the Cauchy problem

dx

dt= iA2x x(0) = x0 (112)

has a unique classical solution given by

x(t) = eitA2x0 t isin R (113)

if x0 isin D(A2) if only x0 isin K2 then (113) is a mild solution of (112) To this end weprove that infin is a regular critical point of A2 thus generalizing a result in [9] for thespecial cases S0 = 0 and S1 = 0 The regularity of infin in fact implies that A2 is the sumof a bounded operator and of an operator which is similar to a self-adjoint operator in aHilbert space (see sect 2) For the operator A1 however infin is in general a singular criticalpoint if H0 is unbounded thus A1 does not generate a group of unitary operators in K1In fact the construction in [17] of the operator A2 from A1 is designed to make infin aregular critical point

In sect 7 for bounded V we show the existence of maximal non-negative and non-positiveinvariant subspaces L+ and Lminus respectively for the definitizable self-adjoint operator A1These subspaces admit so-called angular operator representations eg

L+ =

(x

Kx

) x isin D(K)

with a closed linear operator K in H This yields solutions on the negative and positivehalf-axes of the second-order initial-value problem((

ddτ

+ V

)2

minus H0

)v = 0 v(0) = v0 (114)

which arises from (12) by means of the substitution τ = minusit v(τ) = u(t) The solutionson the positive half-axis are given by

v(τ) = eminusτ(K+V )v0 τ 0

and the admissible set of initial values v0 is the domain D((K + V )2) which in thecase when V = 0 amounts to v0 isin D(H0) The solution on the positive half-axis cor-responds roughly speaking to the spectrum of positive type and the spectrum in theupper (or lower) half-plane whereas the solution on the negative half-axis correspondsto the spectrum of negative type and the spectrum in the upper (or lower) half-planeof A1

Finally in sect 8 we apply the results of the previous sections to the KleinndashGordonequation (11) in R

n We prove that in the space Wminus122 (Rn) it has a unique classical

solution if the initial values ψ0 = ψ(middot 0) and ψ1 = partψ(middot 0)partt belong to W 12 (Rn) and

W122 (Rn) respectively and (minus∆ minus V 2)ψ0 isin W

minus122 (Rn)



2 Preliminaries

21 Notation and definitions from spectral theory

For Hilbert spaces H and Hprime L(HHprime) denotes the space of bounded linear operatorsfrom H to Hprime and we write L(H) if H = Hprime

For a closed linear operator A in a Hilbert space H with domain D(A) we denote byρ(A) σ(A) and σp(A) its resolvent set spectrum and point spectrum or set of eigenvaluesrespectively For λ isin σp(A) the algebraic eigenspace of A at λ is denoted by Lλ(A) Theoperator A is called Fredholm if its kernel is finite dimensional and its range is finitecodimensional (and hence closed see for example [15 Chapter IV sect 51]) The essentialspectrum of A is defined by

σess(A) = λ isin C A minus λ is not Fredholm

An eigenvalue λ0 isin σp(A) is called of finite type if λ0 is isolated (ie a puncturedneighbourhood of λ0 belongs to ρ(A)) and A minus λ0 is Fredholm or equivalently thecorresponding Riesz projection is finite dimensional The set of all eigenvalues of finitetype is called the discrete spectrum of A and denoted by σd(A)

For an analytic operator function F C rarr L(H) the resolvent set and the spectrumof F are defined by

ρ(F ) = λ isin C 0 isin ρ(F (λ)) σ(F ) = C ρ(F )

and the point spectrum or set of eigenvalues of F is the set

σp(F ) = λ isin C 0 isin σp(F (λ))

(see [1430]) The essential spectrum of F is given by

σess(F ) = λ isin C F (λ) is not Fredholm

An eigenvalue λ0 isin σp(F ) is called of finite type if λ0 is isolated (ie a puncturedneighbourhood of λ0 belongs to ρ(F )) and F (λ0) is Fredholm If ρ0 sub C is an openconnected subset of C σess(F ) such that ρ0 cap ρ(F ) = empty then ρ0 cap σ(F ) is at mostcountable with no accumulation point in ρ0 and consists of eigenvalues of finite typeof F and F (middot)minus1 is a finitely meromorphic operator function on ρ0 This means thatF (middot)minus1 is a meromorphic operator function from ρ0 to L(H) for which the coefficients ofthe principal parts of the Laurent expansions at the poles of F (middot)minus1 are all operators offinite rank (cf [15 Corollary XI84 Theorem XVII21]) Conversely if for some openset ρ0 sub C the operator function F (middot)minus1 is finitely meromorphic then ρ0 cap σess(F ) = emptyThe operator function F is called self-adjoint if

F (λ) = F (λ)lowast λ isin C

in particular for λ isin R the values F (λ) are self-adjoint operators The spectrum of aself-adjoint operator function is symmetric with respect to the real axis



22 Self-adjoint operators in Krein spaces

For the definition and simple properties of Krein spaces and the linear operators thereinwe refer the reader to [4725] In the following we recall some of the basic notions AKrein space (K [middot middot]) is a linear space K which is equipped with an (indefinite) innerproduct [middot middot] such that K can be written as

K = G+[]Gminus (21)

where (Gplusmnplusmn[middot middot]) are Hilbert spaces and [] means that the sum of G+ and Gminus is directand [G+Gminus] = 0 The norm topology on a Krein space K is the norm topology ofthe orthogonal sum of the Hilbert spaces Gplusmn in (21) It can be shown that this normtopology is independent of the particular decomposition (21) all the topological notionsin K refer to this norm topology

Krein spaces often arise as follows in a given Hilbert space (G (middot middot)) any boundedself-adjoint operator G in G with 0 isin ρ(G) induces an inner product

[x y] = (Gx y) x y isin G

such that (G [middot middot]) becomes a Krein space In the particular case where G has the addi-tional property G2 = I that is G is the difference of two complementary orthogonalprojections P and Q G = P minus Q with P + Q = I one often writes G = J and in thedecomposition (21) one can choose G+ = PG Gminus = QG

For a closed linear operator A in a Krein space K with dense domain D(A) the (Kreinspace) adjoint A+ of A is the densely defined operator in K with

D(A+) = y isin K [Amiddot y] is a continuous linear functional on D(A)

and[Ax y] = [x A+y] x isin D(A) y isin D(A+)

The operator A is called symmetric in K if A sub A+ and self-adjoint if A = A+ A self-adjoint operator A in a Krein space K may have a non-real spectrum which is alwayssymmetric with respect to the real axis and both the spectrum σ(A) and the resolventset ρ(A) may be empty

An element x isin K is called positive (respectively non-positive neutral etc) if [x x] gt 0(respectively [x x] 0 [x x] = 0 etc) a subspace of K is called positive (respectivelynon-positive neutral etc) if all its non-zero elements are positive (respectively non-positive neutral etc) If for a self-adjoint operator A in a Krein space K with λ0 isinσp(A) all the eigenvectors at λ0 are positive (respectively negative) then λ0 is calledan eigenvalue of positive type (respectively negative type) An eigenvector x0 of A at λ0

that is positive or negative does not have any associated vectors

23 Definitizable operators in Krein spaces

A self-adjoint operator A in a Krein space K is called definitizable if ρ(A) = empty andthere exists a polynomial p such that

[p(A)x x] 0 x isin D(p(A))



The spectrum of a definitizable operator A is real with the possible exception of finitelymany pairs of eigenvalues λ λ which are necessarily zeros of each definitizing polynomialp at such an eigenvalue λ or λ the resolvent of A has a pole of order not greater thanthe order of λ as a zero of the polynomial p The closed linear span of all the algebraiceigenspaces Lλ(A) corresponding to the eigenvalues λ of A in the open upper (or lower)half-plane is a neutral subspace of K The algebraic eigenspaces Lλ(A)Lλ(A) corre-sponding to non-real λ λ isin σp(A) are skewly linked that is to each non-zero x isin Lλ(A)there exists a y isin Lλ(A) such that [x y] = 0 and to each non-zero y isin Lλ(A) there existsan x isin Lλ(A) such that [x y] = 0

If λ isin σp(A) the maximal dimension (up to infin) of a non-negative (non-positiverespectively) subspace of Lλ(A) is denoted by κ+

λ (A) (κminusλ (A) respectively) and the

dimension of the so-called isotropic subspace Lλ(A) cap Lλ(A)[perp] of Lλ(A) is denotedby κo

λ(A) Observe that if for example Lλ(A) is one-dimensional and neutral thenκ+

λ (A) = κminusλ (A) = κo

λ(A) = 1 for a non-real eigenvalue λ of A the number κoλ(A) coin-

cides with the dimension of Lλ(A)A definitizable operator A with definitizing polynomial p has a spectral function with

critical points To introduce it we call a bounded real interval Γ admissible for theoperator A if some definitizing polynomial p of A does not vanish at the end points ofΓ Then for each admissible bounded interval Γ there exists an orthogonal projectionE(Γ ) in K such that the range E(Γ )K is invariant under A the spectrum of the restric-tion A|E(Γ )K is contained in Γ and the real spectrum of the restriction A|(IminusE(Γ ))K iscontained in R Γ

A real spectral point λ isin σ(A) is called of positive type if there exists an admissibleopen interval Γ such that λ isin Γ and (E(Γ )K [middot middot]) is a Hilbert space this is equivalentto the fact that [x x] 0 x isin E(Γ )K (which implies that [x x] gt 0 if x = 0) The set ofall spectral points of positive type of A is denoted by σ+(A) Clearly if (E(Γ )K [middot middot]) isa Hilbert space the restriction A|E(Γ )K has the same spectral properties as a self-adjointoperator in a Hilbert space If a definitizing polynomial p is positive on an admissibleinterval Γ then Γ cap σ(A) consists only of spectral points of positive type Similarly areal point λ isin σ(A) for which there exists an open admissible interval Γ such that λ isin Γ

and (E(Γ )Kminus[middot middot]) is a Hilbert space is called a spectral point of negative type of A theset of all spectral points of negative type of A is denoted by σminus(A) Finally λ0 isin R iscalled a critical point of A if for each admissible open interval Γ with λ0 isin Γ the rangeE(Γ )K contains both positive and negative elements The set of all critical points of A

is denoted by σcrit(A) it is always finite In fact each critical point is a zero of everydefinitizing polynomial p From these definitions it follows that

σ(A) cap R = σ+(A) cup σminus(A) cup σcrit(A)

A real eigenvalue of A with a neutral eigenvector is always a critical point of A Aneigenvalue λ of positive type is a critical point of A if in each neighbourhood of λ thereare spectral points of negative type of A If however the eigenvalue λ of positive type isan isolated spectral point then it is a spectral point of positive type

Similarly infin is called a critical point of A if outside of each compact real intervalthere are spectral points of positive and of negative type of A If infin is a critical point



of the self-adjoint operator A it is called a regular critical point of A if there exists aconstant γ gt 0 such that E(Γ ) γ for all sufficiently large intervals Γ centred at 0otherwise infin is called a singular critical point here middot denotes the operator normcorresponding to the norm in K induced by any of the equivalent decompositions (21)Note that the norm of a self-adjoint projection in a Krein space can be arbitrarily largeIf infin is a regular critical point of A then outside of a sufficiently large compact intervalthe operator A has the same spectral properties as a self-adjoint operator in a Hilbertspace in particular the bounded operators eitA t isin R can be defined and form a groupof unitary operators in K Recall that a unitary operator U in a Krein space K is abounded operator such that

UU+ = U+U = I

The operators occurring in this paper belong to a special class of definitizable opera-tors they are self-adjoint operators A in a Krein space K for which ρ(A) = 0 and thesesquilinear form

[Ax y] x y isin D(A) (22)

has a finite number κ of negative squares recall that the latter means that each subspaceL of K for which

[Ax x] lt 0 x isin L x = 0 (23)

is of dimension less than or equal to κ and for at least one κ-dimensional sub-space L the relation (23) holds In this case the definitizing polynomial is of the formp(λ) = λq(λ)q(λ) with some polynomial q of degree less than or equal to κ Then allthe algebraic eigenspaces corresponding to non-real eigenvalues are finite dimensionalthe positive spectrum of A consists of spectral points of positive type with the possibleexception of a finite number of eigenvalues with a negative or neutral eigenvector andthe negative spectrum of A consists of spectral points of negative type with the possibleexception of a finite number of eigenvalues with a positive or neutral eigenvector Ifadditionally A is boundedly invertible the following equality holds

κ =sum

λisinσp(A)cap(0+infin)

κminusλ (A) +

sumλisinσp(A)cap(minusinfin0)

κ+λ (A) +

sumλisinσp(A)capC+

κoλ(A) (24)

The Krein space K is called a Pontryagin space with negative index κ if in one (andhence in all) decompositions of the form (21) the space Gminus has finite dimension κ Anyself-adjoint operator A in a Pontryagin space is definitizable and hence has a spectralfunction with critical points With the exception of finitely many points the real spectralpoints of A are of positive type the exceptional points are eigenvalues with a negativeor neutral eigenvector and

κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)



3 Operators associated with the abstract KleinndashGordon equationA1 in H oplus H

Let (H (middot middot)) be a Hilbert space with corresponding norm middot let H0 be a uniformly pos-itive self-adjoint operator in H H0 m2 gt 0 and let V be a densely defined symmetricoperator in H

By G1 we denote the orthogonal sum G1 = H oplus H with its inner product

(xxprime)G1 = (x xprime) + (y yprime) x = (x y)T xprime = (xprime yprime)T isin G1

Equipped with the indefinite inner product

[xxprime] = (x yprime) + (y xprime) x = (x y)T xprime = (xprime yprime)T isin G1 (31)

the space K1 = (G1 [middot middot]) becomes a Krein space This is clear since the inner product[middot middot] can be defined by [xxprime] = (Gxxprime)G1 with the Gram operator

G =

(0 I

I 0

)

In G1 = H oplus H with the abstract first-order differential equation (17) we associatethe block operator matrix A1 given by

A1 =

(V I

H0 V

) (32)

With its natural domain D(A1) = (D(V ) cap D(H0)) oplus D(V ) the operator A1 need notbe densely defined nor closable The main assumption here and below is as follows

Assumption 31 D(H120 ) sub D(V )

Since V is closable (with closure denoted by V ) Assumption 31 is satisfied if and onlyif V is H

120 -bounded (see [22 sectsect IV11 IV13]) Since in addition H0 is assumed to

be boundedly invertible Assumption 31 implies that the operator

S = V Hminus120 (33)

is defined on all of H and bounded in H Together with

Hminus120 V sub H

minus120 V lowast sub (V H

minus120 )lowast = Slowast (34)

this shows that Hminus120 V = Slowast

Under Assumption 31 the domain of A1 takes the form

D(A1) =(

x

y

)isin H oplus H x isin D(H0) y isin D(V )

(35)



Theorem 32 If D(H120 ) sub D(V ) then A1 is essentially self-adjoint in the Krein

space K1 Its closure is the self-adjoint operator A1 = (A1)+ given by

D(A1) =(

x

y

)isin H oplus H x isin D(H12

0 ) H120 x + Slowasty isin D(H12

0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)

Proof First we show that

(A1)+ = A1 (36)

In order to prove the inclusion (A1)+ sub A1 let x = (x y)T isin D((A1)+) Then for(A1)+x = u = (u v)T isin G1 = H oplus H we have

[A1xprimex] = [xprimeu] xprime = (xprime yprime)T isin D(A1)

that is

(V xprime + yprime y) + (H0xprime + V yprime x) = (xprime v) + (yprime u) xprime isin D(H0) yprime isin D(V ) (37)

Choosing yprime = 0 we conclude that for all xprime isin D(H0) we have the equality

(V xprime y) + (H0xprime x) = (xprime v)

lArrrArr (V Hminus120 (H12

0 xprime) y) + (H120 (H12

0 xprime) x) = (Hminus120 (H12

0 xprime) v)

lArrrArr (H120 xprime Slowasty) + (H12

0 (H120 xprime) x) = (H12

0 xprime Hminus120 v)

lArrrArr (H120 (H12

0 xprime) x) = (H120 xprime H

minus120 v minus Slowasty)

lArrrArr (H120 wprime x) = (wprime H


with wprime = H120 xprime being an arbitrary element of D(H12

0 ) Hence x isin D(H120 ) and

H120 x = H

minus120 v minus Slowasty

that is

H120 x + Slowasty = H

minus120 v isin D(H12

0 )

and

v = H120 (H12

0 x + Slowasty)

Choosing xprime = 0 in (37) and using the symmetry of V we obtain that for all yprime isin D(V )

(yprime y) + (V yprime x) = (yprime u)

Since x isin D(H120 ) sub D(V ) and D(V ) is dense in H we see that u = V x + y



The inclusion A1 sub (A1)+ follows if we observe that for arbitrary x = (x y)T isin D(A1)and xprime = (xprime yprime)T isin D(A1)

[A1xprimex] = (V xprime y) + (yprime y) + (H0x

prime x) + (V yprime x)

= (H120 xprime Slowasty) + (yprime y) + (H12

0 xprime H120 x) + (V yprime x)

= (H120 xprime H

120 x + Slowasty) + (yprime V x + y)

= [xprime A1x]

By the definition of A1 and A1 and by (36) we have A1 sub A1 = (A1)+ Thus A1 issymmetric in K1 and hence closable Since (A1)+ is closed it follows that

A1 sub (A1)+ = A1

and the theorem is proved if we show that A1 sub A1To this end let (x y)T isin D(A1) Then x isin D(H12

0 ) and y isin H are such that z =H

120 x + Slowasty isin D(H12

0 ) Since D(V ) is dense in H there exists a sequence (yn) sub D(V )with yn rarr y and since S is bounded H

minus120 V yn = Slowastyn rarr Slowasty If we define

xn = z minus Hminus120 V yn isin D(H12

0 ) and xn = Hminus120 xn isin D(H0)

then we have for n rarr infin

xn = Hminus120 z minus H

minus120 H

minus120 V yn rarr H

minus120 (H12

0 x + Slowasty) minus Hminus120 Slowasty = x

H120 xn = xn rarr z minus Slowasty = H

120 x

Since V is H120 -bounded by Assumption 31 this implies that also (V xn) and hence

(V xn + yn) converges Finally

H0xn + V yn = H120 (H12

0 xn + Hminus120 V yn) = H

120 (xn + H

minus120 V yn) = H

120 z

and hence (H0xn + V yn) converges This proves that (x y)T isin D(A1)

In order to ensure that the resolvent set of A1 is non-empty an additional conditionis required in sect 5 In this respect the self-adjoint quadratic operator polynomial


minus120 ) λ isin C (38)

in the Hilbert space H is useful as it reflects the spectral properties of A1 note thataccording to Assumption 31 the values L1(λ) are bounded operators in H

Proposition 33 If D(H120 ) sub D(V ) then for λ isin C

A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H

minus120 (Slowast minus λH

minus120 )

) (39)



Proof The formal equality in (39) follows immediately from formula (38) To provethe equality of the domains we observe that (S minus λH

minus120 )H12

0 = V minus λ on D(H0) Thenthe domain D1 of the product on the right-hand side of (39) is given by

D1 =(

x

y

)isin H oplus H x + H


minus120 )y isin D(H0)

=(

x

y


minus120 Slowasty isin D(H0)

=(

x

y



0 )

which coincides with the domain of A1 by Theorem 32

Proposition 34 Let D(H120 ) sub D(V ) Then ρ(L1) sub ρ(A1) and for λ isin ρ(L1)

(A1 minus λ)minus1

=

(minusH


minus120 )L1(λ)minus1 I

L1(λ)minus1 0

) (I minus(S minus λH

minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1

(I minus(S minus λH

minus120 )Hminus12

0

)

Moreoverσess(A1) sub σess(L1) (310)

Proof The first and the second claim are immediate consequences of the factoriza-tion (39) If λ0 isin σess(L1) then L1(middot)minus1 is a finitely meromorphic operator function ina neighbourhood of λ0 and hence so is (A1 minus middot )minus1 by the second formula for (A1 minus λ)minus1

above This shows that λ0 isin σess(A1)

4 Operators associated with the abstract KleinndashGordon equationA2 in H14 oplus Hminus14

In order to associate a second operator A2 with the abstract KleinndashGordon equation (12)we introduce a scale of Hilbert spaces (Hα middot α) minus1 α 1 induced by the operatorH0 as follows If 0 α 1 we set

Hα = D(Hα0 ) xα = Hα

0 x x isin Hα 0 α 1 (41)

Obviously H0 = H If minus1 α lt 0 then Hα is defined to be the corresponding space withnegative norm (see [6]) which can also be considered as the completion of D(Hα

0 ) = Hwith respect to the norm

xα = Hα0 x x isin H minus1 α lt 0



All powers of H0 extend in a natural way to operators between the spaces of the scaleHα minus1 α 1 in particular Hβ

0 isin L(HαHαminusβ) for α β α minus β isin [minus1 1] The dualitybetween Hα and Hminusα is again denoted by (middot middot)


0 y) x isin Hα y isin Hminusα (42)

Let G2 be the orthogonal sum G2 = H14 oplus Hminus14 with its inner product

(xxprime)G2 = (H140 x H

140 xprime) + (Hminus14

0 y Hminus140 yprime)

for x = (x y)T xprime = (xprime yprime)T isin G2 In addition we equip the space G2 with the indefiniteinner product

[xxprime] = (H140 x H

minus140 yprime) + (Hminus14

0 y H140 xprime)

for x = (x y)T xprime = (xprime yprime)T isin G2 that is

[xxprime] = (x yprime) + (y xprime) (43)

the brackets on the right-hand side of (43) denote the duality (42) between H14 andHminus14 and vice versa

The space K2 = (G2 [middot middot]) is a Krein space To see this we observe that Hminus120 is a

unitary mapping from Gminus14 onto G14 H120 is a unitary mapping from G14 onto Gminus14

and that

[xxprime] = (y xprime) + (x yprime) =

((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2

here the operator J in G2 = H14 oplus Hminus14 is given by

J =

(0 H

minus120

H120 0

)(44)

and satisfies J = Jlowast as well as J2 = I (see sect 22)Imposing Assumption 31 that is D(H12

0 ) sub D(V ) we may consider the symmetricoperator V also as an operator in the scale of spaces Hα minus1 α 1

Remark 41 Assumption 31 is equivalent to the boundedness of V regarded as anoperator from H12 to H that is V isin L(H12H) Due to its symmetry V then admits anextension as a bounded operator from H to Hminus12 by interpolation it can also be definedas a bounded operator from Hα into Hαminus12 for all α isin [0 1

2 ] see [39 Chapter IX4Appendix Example 3] All these extensions and restrictions of the operator V originallygiven in H which act between the spaces of the scale Hα are also denoted by V

V isin L(HαHαminus12) α isin [0 12 ] (45)

As a consequence of (45) for the corresponding extensions and restrictions of the oper-ators S = V H

minus120 and the adjoint Slowast = (V H

minus120 )lowast of S in H we have

S = V Hminus120 isin L(Hα) α isin [minus 1

2 0]



and

Slowast = Hminus120 V isin L(Hα) α isin [0 1

2 ]

Note that in sect 3 the operator Slowast had to be written as Slowast = H120 V since there V acts

only within H and thus requires the domain D(V ) sub H observe also that in Hα α = 0the operator Slowast is not the adjoint of S

Under Assumption 31 we now consider the operator A2 in G2 given by

D(A2) =(

x

y

)isin H14 oplus Hminus14 x isin D(H12

0 ) y isin H

V x + y isin H14 H0x + V y isin Hminus14

A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)

Remark 42 The domain of A2 can also be written as

D(A2) =(

x

y

)isin H14 oplus Hminus14 y isin H V x + y isin H14 H0x + V y isin Hminus14

in fact if y isin H then H0x + V y isin Hminus14 automatically implies x isin D(H120 )

In the following we first show that A2 is a symmetric operator in the Krein space K2In the theorem below we give a sufficient condition for the self-adjointness of A2 in K2

Proposition 43 If D(H120 ) sub D(V ) then A2 is symmetric in K2

Proof Let x = (x y)T isin D(A2) sub D(H120 ) oplus H Then according to the formula (43)

for the inner product [middot middot]

[A2xx] = (V x + y y) + (H0x + V y x) = (V x y) + (y y) + (H0x x) + (V y x)

Note that the first two terms are the inner products in H whereas the last two termsdenote the duality between Gminus12 and G12 Since

(V y x) = (Hminus120 V y H

120 x) = (Slowasty H

120 x) = (y SH

120 x) = (y V x) = (V x y)

it follows that [A2xx] isin R

Theorem 44 Suppose that D(H120 ) sub D(V ) Then

ρ(L1) sub ρ(A2)

the operator A2 is self-adjoint in K2 if ρ(L1) = empty and for λ isin ρ(L1)

(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1

(I minus (S minus λH

minus120 )Hminus12

0

)

(47)



For the proof of Theorem 44 we use the following simple lemma

Lemma 45 If A is a symmetric operator in a Krein space K such that there existsa point λ isin C with λ λ isin ρ(A) then A is self-adjoint in K

Proof Let λ isin C with λ λ isin ρ(A) Then λ isin ρ(A) cap ρ(A+) and the symmetry of A

implies that(A+ minus λ)minus1 sup (A minus λ)minus1

Since λ isin ρ(A) the right-hand side is defined on all of K Hence the last inclusion is infact an equality and so A = A+

Proof of Theorem 44 First we prove that ρ(L1) sub ρ(A2) Let λ0 isin ρ(L1) andlet u = (u v)T isin H14 oplus Hminus14 be arbitrary We have to show that there exists a uniqueelement x = (x y)T isin D(A2) sub D(H12

0 ) oplus H such that (A2 minus λ0)x = u or equivalently

(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)

Since V isin L(H12H) we have u minus (V minus λ0)Hminus10 v isin H and thus

y = L1(λ0)minus1(u minus (V minus λ0)Hminus10 v) isin H (410)

x = Hminus10 (v minus (V minus λ0)y) isin D(H12

0 ) (411)

Then by the definition of x (49) holds Relation (48) follows since

(V minus λ0)x + y = (V minus λ0)Hminus10 (v minus (V minus λ0)y) + y

= (V minus λ0)Hminus10 v + (I minus (V minus λ0)Hminus1

0 (V minus λ0))y

= (V minus λ0)Hminus10 v + (I minus (V H

minus120 minus λ0H

minus120 )(Hminus12

0 V minus λ0Hminus120 ))y

= (V minus λ0)Hminus10 v + L1(λ0)y

= u

here we have used the fact that Hminus120 V = Slowast according to Remark 41 This proves

that A2 minus λ0 is surjective In order to show that A2 minus λ0 is injective let x = (x y)T isinD(A2) sub D(H12

0 ) oplus H be such that (A2 minus λ0)x = 0 or equivalently

(V minus λ0)x + y = 0

H0x + (V minus λ0)y = 0

The second relation yields x = minusHminus10 (V minus λ0)y Inserting this into the first relation we

obtain L1(λ0)y = 0 Now y isin H implies that y = 0 and hence also that x = 0Since L1 is a self-adjoint operator function in H its resolvent set ρ(L1) is symmetric

to R Hence ρ(L1) = empty implies that A2 is self-adjoint in K2 by Lemma 45



It remains to prove the representation for (A2 minus λ)minus1 From (410) (411) it followsthat for λ isin ρ(L1)

(A2 minus λ)minus1 =

(minusHminus1

0 (V minus λ)L1(λ)minus1 Hminus10 + Hminus1

0 (V minus λ)L1(λ)minus1(V minus λ)Hminus10

L1(λ)minus1 minusL1(λ)minus1(V minus λ)Hminus10

)

Using the identities

(V minus λ)Hminus10 = (S minus λH

minus120 )Hminus12

0 Hminus10 (V minus λ) = H


minus120 )

we arrive at (47) Finally we observe that the operators


minus120 )Hminus12

0

) H14 oplus Hminus14 rarr H

L1(λ)minus1 H rarr H(minusH


minus120 )

I

) H rarr H14 oplus Hminus14

are bounded and so the right-hand side of (47) is a bounded operator in the spaceG2 = H14 oplus Hminus14

Corollary 46 If D(H120 ) sub D(V ) we have ρ(L1) sub ρ(A1)capρ(A2) and the resolvents

of A1 and A2 coincide on the dense subset K1 cap K2 = H14 oplus H of K1 and K2

(A1 minus λ)minus1x = (A2 minus λ)minus1x x isin K1 cap K2 λ isin ρ(L1) sub ρ(A1) cap ρ(A2) (412)

Proof The claim follows easily from the formulae for the resolvents of A1 and A2 inProposition 34 and Theorem 44

5 Spectral properties of the operators A1 and A2

In this section we investigate the spectral properties of the self-adjoint operators A1 andA2 in the respective Krein spaces K1 and K2

We recall that under Assumption 31 that is D(H120 ) sub D(V ) the operator A1 in

K1 = H oplus H is given by (see Theorem 32)

D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)



Under the assumption that ρ(L1) = empty the operator A2 in K2 = H14 oplus Hminus14 is givenby (see (46) and Remark 42)

D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)

The definitions of A1 and A2 imply that for x = (x y)T isin D(Aj) sub D(H120 ) oplus H

[Ajxx] = y + Sx2 + ((I minus SlowastS)x x) j = 1 2 (51)

with x = H120 x in H Indeed using Sx = V x we obtain

[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)

= y + Sx2 + ((I minus SlowastS)x x)

the proof for A1 is similarRelation (51) shows that the number of negative squares of the Hermitian forms

[Ajxy]xy isin D(Aj) j = 1 2 coincides with the dimension of the spectral subspaceof the self-adjoint operator I minus SlowastS in H corresponding to the negative half-axis Thefollowing assumption is crucial for the remaining part of this paper it guarantees thatthis number is finite

Assumption 51 S = V Hminus120 = S0 + S1 with S0 lt 1 and S1 compact in H

Obviously such a decomposition of S is not unique and the operator S1 can be chosento have some more particular properties

Lemma 52 In Assumption 51 without loss of generality we can suppose thatS1 =

sumni=1(middot wi)vi with vi isin H and wi isin D(H12

0 ) i = 1 n

Proof Since a compact operator is the sum of an operator with arbitrarily smallnorm and an operator of finite rank S1 can be chosen to be of finite rank sayS1 =

sumni=1(middot wi)vi with vi wi isin H i = 1 n Moreover by means of an additive

perturbation of arbitrarily small norm the elements wi can be chosen in the dense sub-set D(H12

0 ) of H

Lemma 53 If D(H120 ) sub D(V ) and S = V H

minus120 can be decomposed as S = S0+S1

with S0 lt 1 and S1 compact then ρ(L1) = empty



Proof According to the assumption on S for micro isin R the operator L1(imicro) can bewritten as

L1(imicro) = (I + micro2Hminus10 )12(F (micro) + K(micro) + iG(micro))(I + micro2Hminus1

0 )12 (52)

with the bounded self-adjoint operators

F (micro) = I minus (I + micro2Hminus10 )minus12S0S

lowast0 (I + micro2Hminus1

0 )minus12

K(micro) = minus(I + micro2Hminus10 )minus12(S0S

lowast1 + S1S

lowast0 + S1S

lowast1 )(I + micro2Hminus1

0 )minus12 (53)

G(micro) = micro(I + micro2Hminus10 )minus12(SH

minus120 + H

minus120 Slowast)(I + micro2Hminus1

0 )minus12

By (52) imicro isin ρ(L1) if and only if 0 isin ρ(F (micro) + K(micro) + iG(micro)) Since S0 lt 1 and0 (I + micro2Hminus1

0 )minus1 I we have F (micro) I minus S0Slowast0 γ with some γ gt 0 for all micro isin R

The self-adjointness of G(micro) implies that 0 isin ρ(F (micro) + iG(micro)) for all micro isin R and theclaim now follows if we show that K(micro) lt γ for micro isin R sufficiently large

To this end we observe that for x isin H

(I + micro2Hminus10 )minus12x2 =

int Hminus10

0

11 + micro2t

d(E(t)x x) rarr 0 micro rarr infin

where E is the spectral function of Hminus10 Hence the rightmost factor in (53) tends to 0

strongly for micro rarr infin The middle factor in (53) is compact since S1 is compact and theleftmost factor is uniformly bounded for all micro Therefore K(micro) rarr 0 for micro rarr infin (seefor example [51 sect 61])

Lemma 54 Suppose that D(H120 ) sub D(V ) and S = V H

minus120 can be decomposed as

S = S0 + S1 with S0 lt 1 and S1 compact Then the number of negative squares ofthe Hermitian form [A1xy] xy isin D(A1) in H1 and of the Hermitian form [A2xy]xy isin D(A2) in H2 is finite it is equal to the number κ of negative eigenvalues ofI minus SlowastS counted with multiplicities

Proof Both claims follow from relation (51) and from the fact that due to theassumption on S

I minus SlowastS = I minus Slowast0S0 + K

with a compact operator K Observe that Lemma 53 implies that ρ(L1) = empty so that A2

is self-adjoint by Theorem 44

In the following we first consider the particular case S lt 1 which means that theoperator I minus SlowastS is uniformly positive Recall that m gt 0 is such that H0 m2

Lemma 55 If D(H120 ) sub D(V ) and S lt 1 then

σ(L1) sub R (minusα α)

where α = (1 minus S)m



Proof In the proof we use the numerical range W (L1) of the operator polynomial L1

in (38) By definition W (L1) consists of all points λ isin C for which there exists an elementx isin H x = 0 such that (L1(λ)x x) = 0 Since 0 isin ρ(L1) we have σ(L1) sub W (L1)(see [30 Theorem 266]) Hence it is sufficient to show that W (L1) is real and doesnot contain points of the interval (minusα α) The first claim follows from the fact that forarbitrary x isin H with x = 1 the quadratic polynomial

(L1(λ)x x) = x2 minus ((Slowast minus λHminus120 )x (Slowast minus λH

minus120 )x)

is positive at λ = 0 since S lt 1 and tends to minusinfin if λ rarr plusmninfin For the second claimusing H

minus120 1m we obtain that for |λ| lt α

|(L1(λ)x x)| x2 minus (Slowast minus λHminus120 )x2

1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0

and hence λ isin W (L1)

The above lemma can be used to obtain information about the essential spectrum ofL1 in the more general situation of Assumption 51



with S0 lt 1 and S1 compact then

σess(L1) sub R (minusα α)

where α = (1 minus S0)m

Proof By the assumption on S the operator function L1 can be written as

L1(λ) = L0(λ) minus K(λ) λ isin C (54)

here the operator function L0 is given by

L0(λ) = I minus (S0 minus λHminus120 )(Slowast

0 minus λHminus120 ) λ isin C (55)

andK(λ) = S1(Slowast

0 minus λHminus120 ) + (S0 minus λH

minus120 )Slowast

1

is compact for all λ isin C since S1 is compact By Lemma 55 applied to L0 we haveσ(L0) sub R (minusα α) Hence σ(L0) has empty interior as a subset of C and C σ(L0)consists of only one component By the proof of Lemma 53 this component containspoints imicro isin ρ(L1) for micro isin R sufficiently large Now [40 Lemma XIII4] shows that

σess(L1) = σess(L0) sub R (minusα α) (56)



Theorem 57 Suppose that D(H120 ) sub D(V ) and that the operator S = V H

minus120

can be decomposed as S = S0 + S1 with S0 lt 1 and S1 compact Let m gt 0 be suchthat H0 m2 and let κ be the number of negative eigenvalues of I minus SlowastS counted withmultiplicities Then the following statements hold

(i) The self-adjoint operator A1 is definitizable in the Krein space K1

(ii) The non-real spectrum of A1 is symmetric to the real axis and consists of at mostκ pairs of eigenvalues λ λ of finite type the algebraic eigenspaces corresponding toλ and λ are isomorphic

(iii) For the essential spectrum of A1 we have with α = (1 minus S0)m

σess(A1) sub R (minusα α)

(iv) If V is H120 -compact then

σess(A1) = λ isin R λ2 isin σess(H0) sub R (minusm m)

(v) If V Hminus120 lt 1 then the operator A1 is uniformly positive in the Krein space K1

and with α = (1 minus V Hminus120 )m

σ(A1) sub R (minusα α)

Corollary 58 If in addition to the assumptions of Theorem 57 1 isin σp(SlowastS) orequivalently 0 isin σp(A1) then

κ =sum

λisinσp(A1)cap(0+infin)

κminusλ (A1) +

sumλisinσp(A1)cap(minusinfin0)

κ+λ (A1) +

sumλisinσp(A1)capC+

κoλ(A1) (57)

(see (24)) As a consequence the following are true

(i) The number of positive eigenvalues of A1 that have a non-positive eigenvector plusthe number of negative eigenvalues of A1 that have a non-negative eigenvector plusthe number of all eigenvalues of A1 in the open upper half-plane C

+ is at most κ

(ii) With the exception of the real eigenvalues in (i) the spectrum of A1 on the positivehalf-axis is of positive type and the spectrum of A1 on the negative half-axis is ofnegative type

(iii) If V Hminus120 lt 1 that is κ = 0 then all the spectrum of A1 on the positive half-

axis is of positive type and all the spectrum of A1 on the negative half-axis is ofnegative type

(iv) If κ 1 there exists at least one eigenvalue with the properties mentioned in (i)and in particular A1 has at least one eigenvalue

Proof of Theorem 57 (i) We have ρ(L1) = empty by Lemma 53 and ρ(L1) sub ρ(A1) byProposition 34 Thus ρ(A1) = empty and hence the claim follows from Lemma 54 and [24Chapter I3] (see sect 23)

(ii) All claims follow from the definitizability of A1 (see (i) and sect 23)



For the proof of the remaining statements we observe that by (ii) σ(A1) has emptyinterior as a subset of C From Proposition 34 it follows that

(L1(λ)minus1x y) =[(A1 minus λ)minus1

(x

0

)

(0y

)] x y isin H λ isin ρ(L1) (58)

This shows that the analytic operator function L1(middot)minus1 on ρ(L1) can be continued ana-lytically to ρ(A1) and hence ρ(A1) sub ρ(L1) Since ρ(L1) sub ρ(A1) by Proposition 34 wearrive at

ρ(A1) = ρ(L1) (59)

The relation (58) also implies that if λ0 isin σess(A1) and hence (A1 minus middot )minus1 is a finitelymeromorphic operator function in a neighbourhood of λ0 then so is L1(middot)minus1 that isλ0 isin σess(L1) Since σess(A1) sub σess(L1) by (310) we obtain

σess(A1) = σess(L1) (510)

(iii) By (510) and Lemma 56 we have

σess(A1) = σess(L1) sub R (minusα α)

(iv) If V is H120 -compact we can choose S0 = 0 and so (54) and (55) yield that

L1(λ) = L0(λ) + K(λ) where K(λ) is compact and L0 is now given by

L0(λ) = I minus λ2Hminus10 λ isin C

Together with (510) and (56) we obtain that

σess(A1) = σess(L1) = σess(L0) = λ isin R λ2 isin σess(H0)

(v) If S = V Hminus120 lt 1 then equality (59) and Lemma 55 show that

σ(A1) = σ(L1) sub R (minusα α)


minus120



(ii) The spectrum essential spectrum and point spectrum of A1 and A2 coincide

σ(A1) = σ(A2) σess(A1) = σess(A2) σp(A1) = σp(A2) (511)

moreover A1 and A2 have the same Jordan chains and their spectral points are ofthe same (positive or negative) type

(iii) The statements (ii)ndash(v) of Theorem 57 continue to hold for A2



Remark 510 Although A1 and A2 have the same spectra there is an essentialdifference in the behaviour of their spectral functions at infin (see sect 6)

Proof of Theorem 59 (i) We have ρ(L1) = empty by Lemma 53 and ρ(L1) sub ρ(A2)by Theorem 44 Thus ρ(A2) = empty and hence the claim follows from Lemma 54 and [24Chapter I3] (see sect 23)

(ii) First we observe that by (59) and Theorem 44 we have

ρ(A1) = ρ(L1) sub ρ(A2) (512)

and that for λ isin ρ(A1) by (412)

[(A1 minus λ)minus1xy] = [(A2 minus λ)minus1xy] xy isin K1 cap K2 = H14 oplus H (513)

If λ0 is an eigenvalue of finite type of A1 that is λ0 isin σd(A1) then it is either an isolatedeigenvalue of A2 or it belongs to ρ(A2) by (512) Then for j = 1 2 the correspondingRiesz projection is

Pλ0j = minus 12πi

intCλ0

(Aj minus z)minus1 dz

where Cλ0 is a closed Jordan curve in ρ(A1) surrounding λ0 and no other point of thespectra of A1 and A2 Its range is the algebraic eigenspace of Aj at λ0 and the Jordanstructure of Aj in λ0 is determined by the coefficients of the principal part of the Laurentseries of (Aj minus λ)minus1 given by

1λ minus λ0

Pλ0j +pjminus1sumk=1

1(λ minus λ0)k+1 Bk

j

where pj isin N cup infin and Bj = (Aj minus λ0)Pλ0j (see [15 Chapter XV2]) Since λ0 wasassumed to be an eigenvalue of finite type of A1 the projection Pλ01 is finite dimensionalp1 is finite and B1 is a finite rank operator By (513) we have

[Ak1Pλ01xy] = [Ak

2Pλ02xy] xy isin K1 cap K2 (514)

Since K1 cap K2 = H14 oplus H is dense in K1 = H oplus H and in K2 = H14 oplus Hminus14 thecoefficients of the principal parts in the Laurent expansions of (A1 minus λ)minus1 and (A2 minus λ)minus1

at λ0 coincide Hence we have shown that

σd(A1) sub σd(A2) (515)

Since A1 and A2 are definitizable we have

ρ(Aj) cup σd(Aj) cup σess(Aj) = C j = 1 2

This together with (512) and (515) implies that σess(A2) sub σess(A1) sub R It remainsto be shown that

σess(A1) sub σess(A2) (516)



Since the essential spectra of A1 and A2 are both real the spectral projections Ej of Aj

can be represented by means of contour integrals over the resolvent as follows (see [24Chapter I3]) for j = 1 2 and a bounded interval Γ sub R which is admissible for Aj andhas end points a b a b consider the operator

Ej(Γ ) = minus 12πi

int prime

CΓ


Here CΓ is the positively oriented rectangle with corners a+iε b+iε bminus iε and aminus iε forε gt 0 so small that CΓ does not surround any non-real spectral points of A1 and A2 andthe prime denotes the Cauchy principal value of the integral at a and b If the end pointsa b of Γ are not eigenvalues of Aj then Ej(Γ ) = Ej(Γ ) if a is an eigenvalue of Aj andb is not an eigenvalue of Aj then Ej(Γ ) = Ej(Γ ) + 1

2Ej(a) and similarly in all othercases for a and b In any case the operators Ej(Γ ) determine the spectral projections ofAj whence

[E1(Γ )xy] = [E2(Γ )xy] xy isin K1 cap K2

In particular dimE1(Γ )(K1 cap K2) = dimE2(Γ )(K1 cap K2) and consequently E1(Γ ) andE2(Γ ) have the same rank which proves (516)

It remains to be shown that the Jordan chains of A1 and A2 coincide If λ0 isin σp(A1)with Jordan chain (xk)r

k=0 sub D(A1) sub D(H120 ) oplus H r isin N cupinfin then with xminus1 = 0

(A1 minus λ0)xk = xkminus1 k = 0 1 r

Hence for xk = (xkyk)T we have yk isin H and

V xk + yk = xkminus1 + λ0xk isin D(H120 ) sub H14

H120 xk + Slowastyk = H

minus120 (ykminus1 + λ0yk) isin D(H12

0 ) sub H14

(517)

and thus H0xk + V yk isin Hminus14 This shows that (xk)rk=0 sub D(A2) and so (xk)r

k=0 is aJordan chain of A2 at λ0 Conversely if (xk)r

k=0 sub D(A2) sub D(H120 ) oplus H is a Jordan

chain of A2 at λ0 then in (517) the element V xk + yk belongs to D(H120 ) sub H and

the element H120 xk + Slowastyk belongs to D(H12

0 ) Thus (xk)rk=0 sub D(A1) and so (xk)r

k=0is a Jordan chain of A1 at λ0

To conclude this section we compare the spectral properties of A1 with those of theoperator A associated with the abstract KleinndashGordon equation in [27] This operatoracts in the space G = H12 oplus H if Assumption 31 holds and if 1 isin ρ(SlowastS) it is definedas

A =

(0 I

H 2V

) D(A) = D(H) oplus D(H12

0 ) (518)

with H = H120 (I minus SlowastS)H12

0 The operator A is related to the operator A1 introducedin sect 3 by the formula

A = WA1Wminus1 (519)



where W acts from G1 = H oplus H to G = H12 oplus H

W =

(I 0V I

) D(W ) = H12 oplus H

It was shown in [27] that the operator A is self-adjoint in K = (G 〈middot middot〉) with innerproduct

〈xxprime〉 = ((I minus SlowastS)H120 x H

120 xprime) + (y yprime) x = (x y)T xprime = (xprime yprime)T isin G

which is a Pontryagin space due to Assumption 51 Note that the negative index of thisPontryagin space equals the number κ of negative eigenvalues of I minus SlowastS counted withmultiplicities (cf Lemma 54) Between the indefinite inner products 〈middot middot〉 of K and [middot middot]of K1 we have the relation (see [27 Proposition 44 (i)])

〈xxprime〉 = [A1Wminus1x Wminus1xprime] x isin WD(A1) xprime isin G (520)

Theorem 511 Suppose that D(H120 ) sub D(V ) that the operator S = V H

minus120 can

be decomposed as S = S0 + S1 with S0 lt 1 and S1 compact and that 1 isin ρ(SlowastS)Then

σ(A) = σ(A1) = σ(A2) σess(A) = σess(A1) = σess(A2) σp(A) = σp(A1) = σp(A2)

Proof By [27 Theorem 52 (iii)] and Theorem 57 (iii) the non-real spectra of A

and A1 consist of finitely many eigenvalues of finite type If λ isin ρ(A) cap ρ(A1) then forwxprime isin G we obtain from (520) with x = (A minus λ)minus1w isin D(A) sub WD(A1) and (519)

〈(A minus λ)minus1wxprime〉 = [A1Wminus1(A minus λ)minus1w Wminus1xprime]

= [A1(A1 minus λ)minus1Wminus1w Wminus1xprime]

= [Wminus1w Wminus1xprime] + λ[(A1 minus λ)minus1Wminus1w Wminus1xprime]

In a similar way as in the proof of Theorem 59 observing that the range of Wminus1 whichis given by D(W ) = H12 oplusH is dense in G1 = HoplusH one can show that all eigenvaluesof finite type of A1 and A and also their essential spectra coincide

The equality of the point spectra of A and A1 follows from (519) if λ is an eigenvalueof A with eigenvector x isin D(A) then Wminus1x isin D(A1) and Wminus1x is an eigenvectorof A1 corresponding to the eigenvalue λ Conversely if λ is an eigenvalue of A1 witheigenvector x isin D(A1) then Wx isin D(A) and Wx is an eigenvector of A correspondingto the eigenvalue λ

The equalities with the various parts of σ(A2) follow from Theorem 59

6 The critical point infin A2 as a generator of a strongly continuousunitary group

Although the spectra of A1 and A2 coincide their spectral functions E1 and E2 behavedifferently at infin if H0 is unbounded This will be proved first for the case V = 0 for



V = 0 satisfying Assumption 51 a perturbation theorem due to Curgus (see [8]) appliesand shows that infin is a regular critical point for A2

The unperturbed operators A10 in G1 = H oplus H and A20 in G2 = H14 oplus Hminus14 aregiven by

A10 =

(0 I

H0 0

) D(A10) = D(H0) oplus H

A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )

In fact if V = 0 then the operators A1 in G1 and A2 in G2 coincide with the operatorsA10 and A20

Lemma 61 If H0 is unbounded then infin is a regular critical point of A20 whereasit is a singular critical point of A10

Proof The resolvents of A10 and A20 are defined for λ isin C such that λ2 isin σ(H0)they are of the form

(Aj0 minus λ)minus1 =

(λ(H0 minus λ2)minus1 (H0 minus λ2)minus1

I + λ2(H0 minus λ2)minus1 λ(H0 minus λ2)minus1

) j = 1 2

Denote the spectral function of H120 in H by E0 and let Γ sub R be a bounded interval

with Γ gt 0 or Γ lt 0 Using the equality

(H0 minus λ2)minus1 = (H120 minus λ)minus1(H12

0 + λ)minus1 =12λ

((H120 minus λ)minus1 minus (H12

0 + λ)minus1)

and observing that (H120 minus middot )minus1 is holomorphic on Γ if Γ lt 0 and (H12

0 + middot )minus1 isholomorphic on Γ if Γ gt 0 we find that the spectral projection Ej(Γ ) of Aj0 in Kj forj = 1 2 is given by

Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)

Hence for elements x = (x0)T isin G1 = H oplus H we have

E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)

If H0 is unbounded the last term does not remain bounded if Γ gt 0 extends to infin andx isin D(H12

0 )On the other hand for every x = (x y)T isin G2 = H14 oplus Hminus14

E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2

and therefore E2(Γ ) 1



Since for the operator A1 already in the unperturbed situation (that is V = 0 andA1 = A10) the point infin is a singular critical point if H0 is unbounded it will remain asingular critical point for large classes of perturbations (eg for bounded perturbations)

For A2 however in the unperturbed situation (that is V = 0 and A2 = A20) thepoint infin is a regular critical point Due to Assumptions 31 and 51 we may applya perturbation result of Curgus (see [8 Corollary 36]) which guarantees that undercertain additive perturbations the critical point infin remains regular

In order to see this we introduce sesquilinear forms h0 and v in the Hilbert spaceG2 = H14 oplus Hminus14 on D(h0) = D(v) = D(H12

0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)

for x = (x1 x2)T y = (y1 y2)T isin D(H120 ) oplus H It is not difficult to see that the form h0

is closed symmetric and uniformly positive Moreover for x isin D(A20) y isin D(h0) wehave

h0(xy) = [A20xy] = (JA20xy)G2 (61)

where J is defined as in (44) [middot middot] is the indefinite inner product of K2 = H14 oplus Hminus14

(see (43)) and (middot middot)G2 is the corresponding Hilbert space inner product

Lemma 62 Let D(H120 ) sub D(V ) and suppose that S = V H

minus120 can be decomposed

as S = S0 + S1 with S0 lt 1 and S1 compact Then

(i) the form v is h0-bounded with h0-bound less than 1

(ii) the form h = h0 + v defined on D(h0) = D(H120 ) oplus H is closed symmetric and

bounded from below

Proof (i) By the assumption on S and Lemma 52 we may choose the operator S1

to be of the form S1 =sumn

i=1(middot wi)vi with vi isin H and wi isin D(H120 ) i = 1 n Then

the operator S1H120 in H is bounded on D(H12

0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax

for x isin D(H120 ) If we set b = S0 lt 1 we obtain that

V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)

that is V is H120 -bounded with relative bound less than 1 It is not difficult to check

(see [21 sect V41]) that (62) implies that there exist constants aprime bprime 0 bprime lt 1 such that

V x2 aprime2x2 + bprime2H120 x2 x isin D(H12

0 ) (63)

Now let x = (x1 x2)T isin D(v) = D(H120 ) oplus H Using (63) and the inequality

H140 x12 = |(H12

0 x1 x1)| mx12



we obtain the estimate

v(xx) = 2 Re(V x1 x2)

2V x1x2

1bprime V x12 + bprimex22

1bprime (a

prime2x12 + bprime2H120 x12) + bprimex22

aprime2

bprime x12 + bprime(H120 x12 + x22)

aprime2

bprimem(H

140 x12 + H

minus140 x22) + bprime(H

120 x12 + x22)

=aprime2

bprimem(xx)G2 + bprime

h0(xx)

(ii) It is easy to see that h is symmetric since h0 and v are also symmetric All otherclaims follow from the perturbation result [21 Theorem VI133]

Theorem 63 Let D(H120 ) sub D(V ) and suppose that S = V H

minus120 can be decom-

posed as S = S0 + S1 with S0 lt 1 and S1 compact Then infin is a regular critical pointof A2

Proof According to Lemma 62 Assumptions (A) and (B) of [9 sect 2] are satisfiedBy (61) and the first representation theorem (see [21 Theorem VI21]) the operatorJA20 is the self-adjoint operator associated with the closed form h0 in H2 Analogously itfollows that JA2 is the self-adjoint operator associated with the perturbed form h = h0+v

in H2 Since A2 is definitizable by Theorem 59 and infin is a regular critical point for A20

by Lemma 61 it is also a regular critical point for A2 by [9 Proposition 21]

Remark 64 Theorem 63 was proved in [9 Theorem 35] for the particular caseswhen either S0 = 0 or S1 = 0

An important consequence of the regularity of the critical point infin is that A2 is thegenerator of a unitary group in K2 and thus we obtain information on the solvability ofthe Cauchy problem for the differential equation

dx

dt= iA2x

A function x R rarr K2 is called a classical solution of this differential equation if

x isin C1(RK2) x(t) isin D(A2)dx

dt(t) = iA2x(t) t isin R



posed as S = S0 + S1 with S0 lt 1 and S1 compact Then the operator A2 is the



infinitesimal generator of a strongly continuous group (eitA2)tisinR of unitary operatorsin K2 If x0 isin D(A2) the Cauchy problem

dx

dt= iA2x x(0) = x0 (64)



Proof The regularity of the critical point infin by Theorem 63 implies that A2 isthe sum of a bounded operator and an operator similar to a self-adjoint operator Asa consequence the operators eitA2 t isin R are defined and form a strongly continuousgroup of unitary operators in K2 The last claim follows in a similar way as correspondingresults for semi-groups (see for example [23 Theorem 11])

Remark 66 If only x0 isin K2 then (65) is the unique mild solution of (64) that is

x isin C(RK2)int t

s

x(τ) dτ isin D(A2) x(t) = x(s) + iA2

int t

s

x(τ) dτ s t isin R

(see the analogous definitions and results for semi-groups eg in [1 sect 12] [2 3112])

7 Semi-groups associated with A1

In this section we restrict ourselves to the case that the symmetric operator V in H iseverywhere defined and hence bounded Then Assumption 31 used in sect 3 is automaticallysatisfied and the operator A1 in G1 = H oplus H defined in (32) is already closed

A1 = A1 =

(V I

H0 V


Moreover Assumption 51 used in sect 5 holds if V can be decomposed as V = V0 +V1 suchthat V0 lt m and V1H

minus120 is compact

Then according to Theorem 57 the operator A1 is definitizable in K1 and the interval(minus(mminusV0) mminusV0) contains only eigenvalues of finite type in particular there existsa real point micro isin ρ(A1)

Lemma 71 Assume that V is bounded and can be decomposed as V = V0 +V1 suchthat V0 lt m and V1H

minus120 is compact Let κ be the number of negative eigenvalues of

I minus SlowastS counted with multiplicities and let micro isin ρ(A1) cap R There then exists a maximalnon-negative subspace L+ sub K1 that is invariant under (A1 minus micro)minus1 and such that

Im σ((A1 minus micro)minus1|L+) 0

The subspace L+ can be chosen so that it contains all the algebraic eigenspaces of A1

corresponding to the eigenvalues in the open upper half-plane all the positive spectralsubspaces and a non-negative eigenvector of A1 at all real eigenvalues of A1 that are not ofnegative type the negative spectral subspaces of A1 are orthogonal to L+ Furthermore

dim L+ cap L[perp]+ κ (71)



Remark 72 For z in a complex neighbourhood of micro the relation

(A1 minus z)minus1 =infinsum

j=0

(z minus micro)j(A1 minus micro)minusjminus1

holds and implies that the subspace L+ from Lemma 71 is also invariant under (A1minusz)minus1Since ρ(A1) is connected an analytic continuation argument shows that this continuesto hold for all z isin ρ(A1)

Proof of Lemma 71 Since (A1 minus micro)minus1 is a bounded definitizable operator theexistence of a maximal non-negative invariant subspace L0

+ for (A1 minus micro)minus1 follows from[24 Satz 32] If λ0 isin C

+ is an eigenvalue of A1 with algebraic eigenspace Lλ0(A1) thentogether with L0

+ the subspace

L1+ = L0

+ cap (Lλ0(A1) + Lλ0(A1))[perp] Lλ0(A1)

is also a maximal non-negative invariant subspace for (A1 minus micro)minus1 and it contains Lλ0(A1)Since A1 has only a finite number of non-real eigenvalues after repeating this argument afinite number of times the new subspace L+ = Ln

+ contains all the algebraic eigenspacesof A1 corresponding to eigenvalues in the open upper half-plane If L+ does not containall positive subspaces of the form E1(Γ )K1 adding such a subspace to L+ would stillgive a non-negative invariant subspace a contradiction to the maximality of L+ In asimilar way it can be shown that it contains non-negative eigenelements of A1 at all realeigenvalues of A1 that are not of negative type Finally the subspace on the left-handside of (71) is neutral with respect to the inner product [A1middot middot] and hence of dimensionless than or equal to κ

Lemma 73 If L+ is a maximal non-negative subspace as in Lemma 71 then(0y

)isin L+ =rArr y = 0 (72)

Proof It is easy to see that the resolvent of A1 admits the representation

(A1 minus z)minus1 =

(minusD(z)minus1(V minus z) D(z)minus1

I + (V minus z)D(z)minus1(V minus z) minus(V minus z)D(z)minus1

) z isin ρ(A1)

where D(z) = H0 minus (V minus z)2 z isin C For any z isin ρ(A1) we have (A1 minus z)minus1L+ sub L+

which implies that (A1 minus z)minus1L[perp]+ sub L[perp]

+ Hence

(A1 minus z)minus1(L+ cap L[perp]+ ) sub L+ cap L[perp]

+ z isin ρ(A1)

that is (A1 minus z)minus1 maps the isotropic subspace of L+ into itself Since an elementx = (0y)T isin L+ as in (72) is neutral in K1 we have x isin L+ cap L[perp]

+ and so

0 = [(A1 minus z)minus1x (A1 minus z)minus1x] = minus2((V minus Re z)D(z)minus1y D(z)minus1y)

Choosing z isin C such that V minus Re z gt 0 we obtain y = 0



Lemma 73 shows that L+ is the graph of a closed linear operator K in H

L+ =(

x

Kx

) x isin D(K)

(73)

Since for x = (x y)T isin K1 we have [xx] = 2 Re(x y) the non-negativity of L+ yieldsthat the operator K is accretive in H that is Re(Kx x) 0 x isin D(K) (see [21Chapter V310]) The fact that the subspace L+ is maximal non-negative implies thatthe operator K in (73) is maximal accretive in H that is it admits no proper accretiveextension

Remark 74 For a general definitizable operator in a Krein space the maximal non-negative invariant subspace L+ is not uniquely determined and so we do not have auniqueness result for L+ in Lemma 71 However if V = 0 uniqueness can be provedand the maximal non-negative invariant subspace is of the form

L+ =(

x

H120 x

) x isin H

which shows that in this case K = H120

In the following we consider the operator K+V which is in general only quasi-accretive(see [21 Chapter V310]) Therefore we define

ν = min

0 infx=0

(V x x)(x x)

0 (74)

Then the operator K+V minusν is maximal accretive in H and thus generates the contractivestrongly continuous semi-group

S(τ) = eminusτ(K+V minusν) τ 0

As a consequence the operator K + V generates the quasi-bounded strongly continuoussemi-group

T (τ) = eminusτ(K+V ) τ 0 (75)

in H such that T (τ) eτν (cf [10 Theorem 31])The next theorem establishes the existence of solutions of the abstract differential

equation (114)

Theorem 75 Suppose that V is bounded and that the operator S = V Hminus120 can

be decomposed as S = S0 + S1 with S0 lt 1 and S1 compact There then exists amaximal accretive operator K in H such that with the semi-group (T (τ))τ0 given byT (τ) = eminusτ(K+V ) τ 0 for any initial value v0 isin D((K + V )2) the function

v(τ) = T (τ)v0 τ 0 (76)

is a classical solution of the Cauchy problem

v(τ) + 2V v(τ) + V 2v(τ) minus H0v(τ) = 0 τ 0 v(0) = v0 (77)



The spectrum of the infinitesimal generator K +V of the semi-group (T (τ))τ0 containsthe eigenvalues of A1 in the open upper half-plane the spectral points of positive typeof A1 and the real eigenvalues of A1 that are not of negative type

Proof By Theorem 57 (ii) the non-real spectrum of A1 consists only of finitely manypoints Thus we can choose β isin ρ(A1) such that Re β lt ν where ν is given by (74)Then according to Lemma 71 and Remark 72 there exists at least one maximal non-negative subspace L+ in K1 such that (A1 minus β)minus1L+ sub L+ and hence

L+ sub (A1 minus β)(L+ cap D(A1)) (78)

According to Lemma 73 and the remarks following its proof there exists a maximalaccretive operator K in H so that L+ is the graph of K that is (73) holds For thefunction v defined in (76) we have v(τ) isin D((K + V )2) since v0 isin D((K + V )2) andthus the derivatives v(τ) v(τ) exist in the norm topology of H

v(τ) = minus(K + V )v(τ) v(τ) = (K + V )2v(τ) τ 0 (79)

We introduce the function

w(τ) = (K + V minus β)v(τ) τ 0 (710)

Then w(τ) isin D(K + V ) = D(K) and (w(τ)Kw(τ))T isin L+ for all τ 0 Accordingto (78) there exists an element (v(τ)Kv(τ))T isin L+ cap D(A1) such that(

w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0

This equation is equivalent to the system

(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)

for all τ 0 Since Re β lt ν and K is maximal accretive we have β isin ρ(K + V ) Thus(711) and (710) yield v(τ) = v(τ) τ 0 Now (711) and (712) imply that

(H0 + (V minus β)K)v(τ) = K(K + V minus β)v(τ) τ 0

or equivalently

H0v(τ) + 2V (K + V )v(τ) minus V 2v(τ) = (K + V )2v(τ) τ 0

From this we obtain (77) using (79)To prove the last claim we first consider a point λ0 that is either an eigenvalue of A1

in the open upper half-plane or a real eigenvalue of A1 that is not of negative type Inboth cases Lemma 71 shows that there exists an eigenvector x0 of A1 at λ0 that belongs



to L+ This means that there exists an x0 isin D(K) = D(K +V ) so that x0 = (x0 Kx0)T

and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)

Comparing the first components we find (K + V )x0 = λ0x0 ie λ0 isin σp(K + V )Finally we consider a spectral point λ0 of positive type of A1 In this case thereexists a bounded admissible open interval Γ sub R such that λ0 isin Γ and E(Γ )K1 isa positive subspace which is contained in L+ by Lemma 71 Then λ0 is an eigenvalueor approximate eigenvalue of the restriction of A1 to E(Γ )K1 hence there exists asequence (xn) sub E(Γ )K1 sub L+ such that xn = 1 and (A1 minus λ0)xn rarr 0 n rarr infin Asabove it is easy to see that with xn = (xn Kxn)T we have lim infnrarrinfin xn gt 0 and(K + V )xn minus λ0xn rarr 0 n rarr infin and hence λ0 isin σ(K + V )

Remark 76 Using the spectral mapping theorem it can be shown that the spectrumof K + V consists exactly of the points described in the last sentence of the theorem

Remark 77 The function v(τ) = T (τ)v0 τ 0 is defined for arbitrary elementsv0 isin H However if H0 is unbounded it does not necessarily have a second derivativeand hence v is only a solution in some weaker sense

Similar considerations as in this section apply to a maximal non-positive invariantsubspace of A1 which leads to a semi-group and also to a solution of the differentialequation (114) for τ 0

8 Application to the KleinndashGordon equation in Rn

In this section we consider the KleinndashGordon equation (11) in Rn Here H = L2(Rn)

with corresponding inner product (middot middot)2 and norm middot2 H0 = minus∆+m2 and V = eq is themaximal multiplication operator by the real-valued measurable function eq R

n rarr RIn the following we formulate necessary and sufficient conditions for Assumptions 31and 51 and we apply the results of the previous sections to the KleinndashGordon equationin R

nMany different sufficient conditions for the relative boundedness and for the relative

compactness of a multiplication operator with respect to H120 = (minus∆ + m2)12 have been

established (see for example [223943] and the more specialized references therein)Examples considered in [27 sect 6] include Rollnik potentials which are (minus∆ + m2)12-bounded and potentials belonging to Lp(Rn) with n p lt infin which are (minus∆+m2)12-compact The most general description in terms of necessary and sufficient conditionswhich we present here has been given by Mazprimeya and Shaposhnikova in [3133]

81 Assumptions 31 and 51

It is well known (see [44 sectsect 131 132]) that for H0 = minus∆ + m2 the spaces Hα =D(Hα

0 ) α isin [0 1] introduced in (41) are the Sobolev spaces of order 2α associated with



L2(Rn) (see the definition given below)

Hα = W 2α2 (Rn) α isin [0 1]

Hence Assumption 31 which reads H12 = D(H120 ) sub D(V ) now becomes

Assumption 81 W 12 (Rn) sub D(V )

This is equivalent to V isin L(W 12 (Rn) L2(Rn)) or to the fact that V is (minus∆ + m2)12-

bounded the latter means that there exist constants a b 0 such that

V u2 au2 + b(minus∆ + m2)12u2 u isin W 12 (Rn) (81)

Therefore Assumption 51 (and hence Assumption 31) is satisfied if the restriction of V

to W 12 (Rn) can be decomposed as follows

Assumption 82 V = V0 + V1 such that V0 is (minus∆ + m2)12-bounded and satisfies(81) with am + b lt 1 and V1 is (minus∆ + m2)12-compact

Before we formulate abstract necessary and sufficient conditions for Assumptions 81and 82 we consider an example for which both assumptions may be checked directlyusing Hardyrsquos inequality and Sobolevrsquos embedding theorems (see [27 Theorem 61Proposition 64 and Example 65])

Example 83 Let n 3 Assumption 82 is satisfied for

V (x) =γ

|x| + V1(x) x isin Rn 0

with γ isin R |γ| lt (n minus 2)2 and V1 isin Lp(Rn) n p lt infin

Instead of the Coulomb part V0(x) = γ|x| we could also assume that V0 is boundedV0 isin Linfin(Rn) with V0infin lt m or that V 2

0 is a so-called Rollnik potential (see [3943])with Rollnik norm V 2

0 R lt 4π (see [27 Theorem 62])Note that the admission of the relatively compact part V1 of V which is not subject

to any relative norm bound may give rise to complex eigenvalues even if V is a boundedpotential This was observed in [42] for potentials represented by a sufficiently deep well

In general the property that V0 is (minus∆+m2)12-bounded means that V0 belongs to thespace M(W 1

2 (Rn) L2(Rn)) of bounded multipliers from the Sobolev space W 12 (Rn) into

L2(Rn) the property that V1 is (minus∆+m2)12-compact means that V1 belongs to the spaceM(W 1

2 (Rn) L2(Rn)) of compact multipliers from W 12 (Rn) into L2(Rn) (see [33]) Neces-

sary and sufficient conditions for functions V to belong to a space M(Wm2 (Rn) W l

2(Rn))

of bounded multipliers or the corresponding space of compact multipliers have beenestablished by Mazprimeya and Shaposhnikova for integer m and l in [31] and for fractionalm and l in [32] (see [33]) In view of Assumption 31 we restrict ourselves to the casel = 0 In the following we introduce all necessary definitions to formulate these criteriain Theorem 84 below



If S1 and S2 are Banach spaces of functions on Rn then a function γ S1 rarr S2 is

called a bounded multiplier from S1 into S2 γ isin M(S1S2) if

γM(S1S2) = supγuS2 u isin S1 uS1 = 1 lt infin

With any Banach space S of functions on Rn we associate the space

Sloc = u Rn rarr C for all η isin Cinfin

0 (Rn) and ηu isin S

For s gt 0 we denote by [s] and s the integer and fractional part respectively of sie s = [s] + s with [s] isin N and 0 s lt 1 For k isin N we denote by nablak the gradientof order k that is

nablak =(

partk

partxα11 middot middot middot partxαn

n

)α1+middotmiddotmiddot+αn=k

For s gt 0 and 1 p lt infin the fractional derivative Dps of order s in Lp(Rn) of a functionu on R

n is given by

(Dpsu)(x) =( int

Rn

|(nabla[s]u)(x + h) minus (nabla[s]u)(x)||h|nminusps dh

)1p

The fractional Sobolev space W sp (Rn) is defined as the closure of Cinfin

0 (Rn) with respectto the norm

ups = Dspup + up u isin W sp (Rn) (82)

henceforth middot p denotes the standard norm in Lp(Rn) For integer s = k isin N the spaceW s

p (Rn) coincides with the classical Sobolev space

W kp (Rn) = u isin Lp(Rn) nablaju isin Lp(Rn) j = 1 2 k

and the norm (82) is equivalent to

upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)

Finally for a compact subset e sub Rn we define the (p s)-capacity of e by

cap(e W sp (Rn)) = infups u isin Cinfin

0 (Rn) u|e 1

In the following if ω varies in some set Ω and a b depend on ω we write a(ω) sim b(ω) ifthere exist constants c1 c2 gt 0 such that c1b(ω) a(ω) c2b(ω) for all ω isin Ω

Theorem 84 Let V Rn rarr C be a measurable function m isin N and p isin (1infin)

Then V isin M(Wmp (Rn) Lp(Rn)) (the space of bounded multipliers) if and only if there

exists a constant c gt 0 such that for any compact subset e sub Rn

V epp =

inte

|V (x)|p dx c cap(e Wmp (Rn))



and

V M(W mp (Rn)Lp(Rn)) sim sup

esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p

Furthermore V isin M(Wmp (Rn) Lp(Rn)) (the space of compact multipliers) if and only

if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0

The proof of this theorem may be found in [33 sect 214 and Lemma 2221]

82 Application of Theorems 57 59 and 65

If the potential V satisfies Assumption 82 (and hence Assumptions 31 and 51) thenall results of the previous sections apply to the KleinndashGordon equation in R

n and weobtain the following statements

Recall that by Assumption 81 the operator V maps the space W k2 (Rn) boundedly

onto W kminus12 (Rn) for all k isin [0 1] (see Remark 41 (45))

Theorem 85 Suppose that W 12 (Rn) sub D(V ) and that V = V0 + V1 where V0 is

(minus∆+m2)12-bounded satisfying (81) with am+b lt 1 and V1 is (minus∆+m2)12-compact

(i) The operator A2 given by

D(A2) =(

x

y

)isin W

122 (Rn) oplus W

minus122 (Rn) x isin W 1

2 (Rn) y isin L2(Rn)

V x + y isin W122 (Rn) (minus∆ + m2)x + V y isin W

minus122 (Rn)

A2

(x

y

)=

(V x + y

(minus∆ + m2)x + V y

)

is a self-adjoint and definitizable operator in the Krein space given by K2 =W

122 (Rn) oplus W

minus122 (Rn) with indefinite inner product

[xxprime] = ((minus∆+m2)14x (minus∆+m2)minus14yprime)2+((minus∆+m2)minus14y (minus∆+m2)14xprime)2

for x = (x y)T xprime = (xprime yprime)T isin W122 (Rn) oplus W

minus122 (Rn)

(ii) The non-real spectrum of A2 is symmetric to the real axis and consists of at mostfinitely many pairs of eigenvalues λ λ of finite type the algebraic eigenspaces cor-responding to λ and λ are isomorphic There are no complex eigenvalues if

V (minus∆ + m2)minus12 lt 1

(iii) The essential spectrum of A2 is real and has a gap around 0 more precisely

σess(A2) sub R (minusα α) α =(

1 minus(

a

m+ b

))m



(iv) The operator A2 is generates a strongly continuous group (eitA2)tisinR of unitaryoperators in the Krein space K2 = W

122 (Rn) oplus W

minus122 (Rn)

(v) For every initial value x0 isin D(A2) the Cauchy problem

dx

dt= iA2x x(0) = x0

has a unique classical solution x isin C1(R W122 (Rn) oplus W

minus122 (Rn)) given by x(t) =

eitA2x0 t isin R

The Cauchy problem for A2 is equivalent to an initial-value problem for the KleinndashGordon equation (11) This leads to the following consequence of Theorem 85 (v)

Theorem 86 Suppose that the potential V satisfies the assumptions of Theorem 85Then the initial-value problem((

part

parttminus ieq

)2

minus ∆ + m2)

ψ = 0 ψ(middot 0) = ψ0partψ

partt(middot 0) = ψ1 (83)

in the space Wminus122 (Rn) has a unique classical solution ψs if ψ0 isin W 1

2 (Rn) ψ1 isinW

122 (Rn) and (minus∆ minus V 2)ψ0 isin W

minus122 (Rn) and the function t rarr ψs(middot t) belongs to

C1(R W122 (Rn)) cap C2(R W

minus122 (Rn))

Proof The Cauchy problem for A2 arises from (83) by means of the substitution

x(t) = ψ(middot t) y(t) =(

minus ipart

parttminus V

)ψ(middot t) t isin R

hence the initial value for the Cauchy problem for A2 is given by

x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ

partt(middot 0) minus V ψ(middot 0)

⎞⎠ =

(ψ0

minusiψ1 minus V ψ0

)

Since V maps W 12 (Rn) boundedly in L2(Rn) by Assumption 81 the assumptions on

ψ0 and ψ1 guarantee that x0 isin D(A2) Hence Theorem 85 (v) yields a unique solutionx = (x y)T isin C1(R W

122 (Rn) oplus W

minus122 (Rn)) satisfying the equations

dx

dt= i(V x + y) in W

122 (Rn)

dy

dt= i((minus∆ + m2)x + V y) in W

minus122 (Rn)

Differentiating the first equation and using the second one we obtain that x isinC1(R W

122 (Rn)) cap C2(R W

minus122 (Rn)) and that x satisfies (83) in W

minus122 (Rn)

Remark 87 If only ψ0 isin W122 (Rn) and ψ1 isin W

minus122 (Rn) then x0 isin W

122 (Rn) oplus

Wminus122 (Rn) In this case we only obtain mild solutions of the Cauchy problem for A2

(see Remark 66) and thus solutions of (83) in some weaker sense



If the potential V is (minus∆ + m2)12-compact a similar result was proved in [18 sect 5]also using the regularity of the critical point infin of A2

The above theorem should also be compared with a corresponding result which canbe obtained using the operator A introduced in [27] (see sect 5) Since A is a self-adjointoperator in the Pontryagin space K = H12 oplus H = W 1

2 (Rn) oplus L2(Rn) it generates agroup (eitA)tisinR of unitary operators in this space By means of the substitution

x(t) = ψ(middot t) y(t) = minusipartψ

partt(middot t) t isin R

one can prove an existence and uniqueness result for classical solutions of (83) in L2(Rn)Here stronger assumptions on the initial values have to be imposed which ensure that

x(0) = (ψ0 minus iψ1)T isin D(A) = D(H) oplus D(H120 ) sub W 2

2 (Rn) oplus W 12 (Rn)

(H = H120 (I minus SlowastS)H12

0 being the operator associated with the differential expressionminus∆ + V 2)

Remark 88 Suppose that the potential V satisfies the assumptions of Theorem 85and that 1 isin ρ(SlowastS) Then the initial problem (83) in L2(Rn) has a unique classicalsolution ψs if ψ0 isin D(H) sub W 2

2 (Rn) ψ1 isin W 12 (Rn) and the function t rarr ψs(middot t) belongs

to C1(R W 12 (Rn)) cap C2(R L2(Rn))

This shows that for smooth initial values as in Remark 88 the operator A studiedin [27] gives classical solutions of the KleinndashGordon equation (83) in L2(Rn) for lesssmooth initial values as in Theorem 86 the operator A2 studied in the present paperstill gives classical solutions of the KleinndashGordon equation but only in W

minus122 (Rn)

Acknowledgements HL and CT gratefully acknowledge the support of DeutscheForschungsgemeinschaft under Grant no TR3686-1 We are grateful to our late col-league Dr Peter Jonas for valuable comments which led to various improvements of thispaper he died on 18 July 2007 shortly before the final version of the manuscript wascompleted

References

1 W Arendt Semigroups and evolution equations functional calculus regularity andkernel estimates in Evolutionary equations Handbook of Differential Equations Volume Ipp 1ndash85 (North-Holland Amsterdam 2004)

2 W Arendt C J K Batty M Hieber and F Neubrander Vector-valued Laplacetransforms and Cauchy problems Monographs in Mathematics Volume 96 (BirkhauserBasel 2001)

3 B Asmuszlig Zur Quantentheorie geladener Klein-Gordon Teilchen in auszligeren FeldernPhD thesis Hagen Germany (1990)

4 T Ya Azizov and I S Iokhvidov Linear operators in spaces with an indefinite metricPure and Applied Mathematics (Wiley 1989)

5 A Bachelot Superradiance and scattering of the charged KleinndashGordon field by a step-like electrostatic potential J Math Pures Appl 83 (2004) 1179ndash1239



6 Y M Berezansky Z G Sheftel and G F Us Functional analysis Volume IIOperator Theory Advances and Applications Volume 86 (Birkhauser Basel 1996)

7 J Bognar Indefinite inner product spaces Ergebnisse der Mathematik und ihrer Grenz-gebiete Volume 78 (Springer 1974)

8 B Curgus On the regularity of the critical point infinity of definitizable operators IntegEqns Operat Theory 8 (1985) 462ndash488

9 B Curgus and B Najman Quasi-uniformly positive operators in Kreın space in Oper-ator theory and boundary eigenvalue problems Operator Theory Advances and Applica-tions Volume 80 pp 90ndash99 (Birkhauser Basel 1995)

10 E B Davies One-parameter semigroups London Mathematical Society MonographsVolume 15 (Academic London 1980)

11 K-J Eckardt On the existence of wave operators for the KleinndashGordon equationManuscr Math 18 (1976) 43ndash55

12 K-J Eckardt Scattering theory for the KleinndashGordon equation Funct Approx Com-ment Math 8 (1980) 13ndash42

13 H Feshbach and F Villars Elementary relativistic wave mechanics of spin 0 andspin 1

2 particles Rev Mod Phys 30 (1958) 24ndash4514 I C Gohberg and M G Kreın Introduction to the theory of linear nonselfadjoint

operators Translations of Mathematical Monographs Volume 18 (American MathematicalSociety Providence RI 1969)

15 I Gohberg S Goldberg and M A Kaashoek Classes of linear operators Volume IOperator Theory Advances and Applications Volume 49 (Birkhauser Basel 1990)

16 P Jonas On local wave operators for definitizable operators in Krein space and on apaper by T Kako preprint P-4679 (Zentralinstitut fur Mathematik und Mechanik derAdW DDR Berlin 1979)

17 P Jonas On a problem of the perturbation theory of selfadjoint operators in Kreınspaces J Operat Theory 25 (1991) 183ndash211

18 P Jonas On the spectral theory of operators associated with perturbed KleinndashGordonand wave type equations J Operat Theory 29 (1993) 207ndash224

19 P Jonas On bounded perturbations of operators of KleinndashGordon type Glas Mat SerIII 35 (2000) 59ndash74

20 T Kako Spectral and scattering theory for the J-selfadjoint operators associated withthe perturbed KleinndashGordon type equations J Fac Sci Univ Tokyo (1) A23 (1976)199ndash221

21 T Kato Perturbation theory for linear operators Die Grundlehren der mathematischenWissenschaften Volume 132 (Springer 1966)

22 T Kato A short introduction to perturbation theory for linear operators (Springer 1982)23 S G Kreın Linear differential equations in Banach space Translations of Mathematical

Monographs Volume 29 (American Mathematical Society Providence RI 1971)24 H Langer Invariante Teilraume definisierbarer J-selbstadjungierter Operatoren An-

nales Acad Sci Fenn A475 (1971) 1ndash2325 H Langer Spectral functions of definitizable operators in Kreın spaces in Functional

analysis Lecture Notes in Mathematics Volume 948 pp 1ndash46 (Springer 1982)26 H Langer and B Najman A Krein space approach to the KleinndashGordon equation

unpublished manuscript (1996)27 H Langer B Najman and C Tretter Spectral theory of the KleinndashGordon equation

in Pontryagin spaces Commun Math Phys 267 (2006) 159ndash18028 L-E Lundberg Relativistic quantum theory for charged spinless particles in external

vector fields Commun Math Phys 31 (1973) 295ndash31629 L-E Lundberg Spectral and scattering theory for the KleinndashGordon equation Com-

mun Math Phys 31 (1973) 243ndash257



30 A S Markus Introduction to the spectral theory of polynomial operator pencils Trans-lations of Mathematical Monographs Volume 71 (American Mathematical Society Prov-idence RI 1988)

31 V G Mazprimeya and T O Shaposhnikova Multipliers in pairs of spaces of potentials

Math Nachr 99 (1980) 363ndash37932 V G Maz

primeya and T O Shaposhnikova Multipliers in pairs of spaces of differentiable

functions Trudy Mosk Mat Obshc 43 (1981) 37ndash80 (in Russian)33 V G Maz

primeya and T O Shaposhnikova Theory of multipliers in spaces of differen-

tiable functions Monographs and Studies in Mathematics Volume 23 (Pitman BostonMA 1985)

34 B Najman Solution of a differential equation in a scale of spaces Glas Mat Ser III14 (1979) 119ndash127

35 B Najman Spectral properties of the operators of KleinndashGordon type Glas Mat SerIII 15 (1980) 97ndash112

36 B Najman Localization of the critical points of KleinndashGordon type operators MathNachr 99 (1980) 33ndash42

37 B Najman Eigenvalues of the KleinndashGordon equation Proc Edinb Math Soc 26(1983) 181ndash190

38 J Palmer Symplectic groups and the KleinndashGordon field J Funct Analysis 27 (1978)308ndash336

39 M Reed and B Simon Methods of modern mathematical physics II Fourier analysisself-adjointness (Academic 1975)

40 M Reed and B Simon Methods of modern mathematical physics IV Analysis of oper-ators (AcademicHarcourt Brace 1978)

41 M Schechter The KleinndashGordon equation and scattering theory Annals Phys 101(1976) 601ndash609

42 L I Schiff H Snyder and J Weinberg On the existence of stationary states of themesotron field Phys Rev 57 (1940) 315ndash318

43 B Simon Quantum mechanics for Hamiltonians defined as quadratic forms PrincetonSeries in Physics (Princeton University Press 1971)

44 H Triebel Theory of function spaces II Monographs in Mathematics Volume 84(Birkhauser Basel 1992)

45 K Veselic A spectral theory for the KleinndashGordon equation with an external electro-static potential Nucl Phys A147 (1970) 215ndash224

46 K Veselic On spectral properties of a class of J-selfadjoint operators I Glas MatSer III 7 (1972) 229ndash248

47 K Veselic On spectral properties of a class of J-selfadjoint operators II Glas MatSer III 7 (1972) 249ndash254

48 K Veselic A spectral theory of the KleinndashGordon equation involving a homogeneouselectric field J Operat Theory 25 (1991) 319ndash330

49 R Weder Selfadjointness and invariance of the essential spectrum for the KleinndashGordonequation Helv Phys Acta 50 (1977) 105ndash115

50 R Weder Scattering theory for the KleinndashGordon equation J Funct Analysis 27(1978) 100ndash117

51 J Weidmann Linear operators in Hilbert spaces Graduate Texts in Mathematics Vol-ume 68 (Springer 1980)



self-adjoint operator H0 in a Hilbert space H and the operator of multiplication withthe function eq by a symmetric operator V in H we obtain the abstract KleinndashGordonequation ((

ddt

minus iV)2

+ H0

)u = 0 (12)

where u is a function of t isin R with values in H Equation (12) can be transformed into afirst-order differential equation for a vector function x in an appropriate space formallygiven by H oplus H and a linear operator A therein

dx

dt= iAx (13)

However this is in general not possible with a self-adjoint operator A in a Hilbert spaceIn the literature two inner products have been associated with the KleinndashGordon equa-

tion (11) one of them representing the charge and the other one representing the energyof the particle The energy inner product is used in numerous papers to study the spec-tral and scattering properties of (11) see for example [511121620282935ndash3741424950] The charge inner product has been suggested by the physics literature(see [13]) it was first used in the pioneering work of Veselic [45ndash47] and subsequentlyin the papers [343848] the unpublished manuscriptlowast [26] as well as in [917ndash19] Itis also called the number norm since it is related to the number operator in the theoryof second quantization (see [3])

The reason for the preference of the energy inner product 〈middot middot〉 may be that it is positivedefinite and generates a Hilbert space if the potential V is small with respect to H

120

this can be seen from its formal definition

〈xxprime〉 =

((H0 minus V 2 0

0 I

)xxprime

)= ((H0 minus V 2)x xprime) + (y yprime) (14)

for suitable elements x = (x y)T xprime = (xprime yprime)T of H oplus H where (middot middot) denotes the scalarproduct in H The charge inner product [middot middot] however is always indefinite it is definedon elements x = (x y)T xprime = (xprime yprime)T of H oplus H by a relation of the form

[xxprime] =

((0 I

I 0

)xxprime

)= (x yprime) + (y xprime) (15)

Therefore it is negative on an infinite-dimensional subspace (if H is infinite dimensional)and hence leads to a so-called Krein space At first glance these indefinite structures seemto be less convenient from the mathematical point of view However they allow deeperinsight into the spectral properties of the KleinndashGordon equation eg by providing aclassification of the points of the spectrum into points of positive negative or neutraltype and sufficient conditions for the existence of corresponding strongly continuousgroups of operators which are unitary with respect to the indefinite inner product (15)

lowast This manuscript dating back to the late 1980s was the starting point for the present paper andalso for the papers [1827]




x = u y =(

minus iddt

minus V

)u (16)


dx

dt= i

(V I

H0 V

)x (17)





V I

H0 V

)xxprime

]=

((H0 V

V I

)xxprime

) (18)


















minus120 lt 1





x = u y = minusidu

dt (110)


dx

dt= i

(0 I

H0 minus V 2 2V

)x (111)









dx

dt= iA2x x(0) = x0 (112)





L+ =

(x

Kx

) x isin D(K)


ddτ

+ V

)2

minus H0

)v = 0 v(0) = v0 (114)








minus122 (Rn)



2 Preliminaries



















K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































x = u y =(

minus iddt

minus V

)u (16)


dx

dt= i

(V I

H0 V

)x (17)





V I

H0 V

)xxprime

]=

((H0 V

V I

)xxprime

) (18)


















minus120 lt 1





x = u y = minusidu

dt (110)


dx

dt= i

(0 I

H0 minus V 2 2V

)x (111)









dx

dt= iA2x x(0) = x0 (112)





L+ =

(x

Kx

) x isin D(K)


ddτ

+ V

)2

minus H0

)v = 0 v(0) = v0 (114)








minus122 (Rn)



2 Preliminaries



















K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References






































































minus120 lt 1





x = u y = minusidu

dt (110)


dx

dt= i

(0 I

H0 minus V 2 2V

)x (111)









dx

dt= iA2x x(0) = x0 (112)





L+ =

(x

Kx

) x isin D(K)


ddτ

+ V

)2

minus H0

)v = 0 v(0) = v0 (114)








minus122 (Rn)



2 Preliminaries



















K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































dx

dt= iA2x x(0) = x0 (112)





L+ =

(x

Kx

) x isin D(K)


ddτ

+ V

)2

minus H0

)v = 0 v(0) = v0 (114)








minus122 (Rn)



2 Preliminaries



















K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References






























































2 Preliminaries



















K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































K = G+[]Gminus (21)


































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
















































































UU+ = U+U = I




[Ax x] lt 0 x isin L x = 0 (23)


κ =sum


κminusλ (A) +


κ+λ (A) +


κoλ(A) (24)


κ =sum

λisinσp(A)capR

κminusλ (A) +


κoλ(A)










G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References





































































G =

(0 I

I 0

)


A1 =

(V I

H0 V

) (32)








Hminus120 V sub H





D(A1) =(

x

y


(35)





D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)


(A1)+ = A1 (36)



that is





0 xprime) y) + (H120 (H12


0 xprime) v)


0 (H120 xprime) x) = (H12


lArrrArr (H120 (H12







H120 x = H


that is



0 )

and

v = H120 (H12

0 x + Slowasty)











= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References



































































= (H120 xprime H


= [xprime A1x]


A1 sub (A1)+ = A1










minus120 H


minus120 (H12



120 x



H0xn + V yn = H120 (H12


120 (xn + H

minus120 V yn) = H

120 z







A1 minus λ =

(I (S minus λH

minus120 )H12

0

0 H0

) (0 L1(λ)I H


minus120 )

) (39)




minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































minus120 )H12


D1 =(

x

y




=(

x

y



=(

x

y



0 )



(A1 minus λ)minus1

=

(minusH



L1(λ)minus1 0


minus120 )Hminus12

0

0 Hminus10

)

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)







0 x x isin Hα 0 α 1 (41)

















0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References









































































0 y H140 xprime)






and that


((H

minus120 y

H120 x

)

(xprime

yprime

))G2

= (Jxxprime)G2


J =

(0 H

minus120

H120 0

)(44)










2 0]



and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References






























































and


2 ]




D(A2) =(

x

y


0 ) y isin H


A2

(x

y

)=

(V x + y

H0x + V y

)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(46)


D(A2) =(

x

y











120 x) = (y SH

120 x) = (y V x) = (V x y)



ρ(L1) sub ρ(A2)


(A2 minus λ)minus1

=

(0 Hminus1

0

0 0

)+

(minusH


minus120 )

I

)L1(λ)minus1


minus120 )Hminus12

0

)

(47)










(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References





































































(V minus λ0)x + y = u (48)

H0x + (V minus λ0)y = v (49)




0 ) (411)




0 (V minus λ0))y


minus120 minus λ0H

minus120 )(Hminus12



= u













(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































(minusHminus1




)



minus120 )Hminus12



minus120 )



minus120 )Hminus12

0




minus120 )

I











D(A1) =(

x

y



0 )

A1

(x

y

)=

(V x + y

H120 (H12

0 x + Slowasty)

)




D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































D(A2) =(

x

y


0 ) y isin H


=(

x

y


A2

(x

y

)=

(V x + y

H0x + V y

)




[A2xx] = (V x + y y) + (H0x + V y x)

= H120 x2 + y2 + (V x y) + (y V x)

= H120 x2 + y2 + (Sx y) + (y Sx)








0 ) i = 1 n




0 ) of H








0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































0 )12 (52)




0 )minus12


lowast1 + S1S

lowast0 + S1S


0 )minus12 (53)


minus120 + H


0 )minus12






int Hminus10

0

11 + micro2t




















minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References

































































minus120 )x)




1 minus(

S +|λ|m

)2

gt 1 minus(

S +α

m

)2

0















minus120 )Slowast

1






minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































minus120












κ =sum


κminusλ (A1) +


κ+λ (A1) +


κoλ(A1) (57)



+ is at most κ











(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































(x

0

)

(0y



ρ(A1) = ρ(L1) (59)













minus120












ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References

































































ρ(A1) = ρ(L1) sub ρ(A2) (512)




Pλ0j = minus 12πi

intCλ0



1λ minus λ0



j


[Ak1Pλ01xy] = [Ak




σd(A1) sub σd(A2) (515)










int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References

































































int prime

CΓ













0 ) sub H14

(517)









A =

(0 I

H 2V


0 ) (518)







W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































W =

(I 0V I








minus120 can

















A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































A10 =

(0 I

H0 0


A20 =

(0 I

H0 0

) D(A20) = D(H34

0 ) oplus D(H140 )







) j = 1 2




0 + λ)minus1 =12λ


0 + λ)minus1)



Ej(Γ ) = plusmn12

(E0(Γ ) H

minus120 E0(Γ )

H120 E0(Γ ) E0(Γ )

)


E1(Γ )x2G1

= 14 (E0(Γ )x2 + H

120 E0(Γ )x2)



E2(Γ )x2G2

= 14H

140 (E0(Γ )x + H

minus120 E0(Γ )y)2

+ 14H

minus140 (H12

0 E0(Γ )x + E0(Γ )y)2

H140 x2 + H

minus140 y2

= x2G2







0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References

































































0 ) oplus H by

h0(xy) = (H120 x1 H

120 y1) + (x2 y2)

v(xy) = (V x1 y2) + (x2 V y1)



h0(xy) = [A20xy] = (JA20xy)G2 (61)








bounded from below





0 )

S1H120 x

nsumi=1

|(x H120 wi)| vi

( nsumi=1

H120 wi vi

)x = ax


V x = (S0 + S1)H120 x ax + bH

120 x x isin D(H12

0 ) (62)




0 ) (63)


H140 x12 = |(H12

0 x1 x1)| mx12




v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































v(xx) = 2 Re(V x1 x2)

2V x1x2


1bprime (a


aprime2


aprime2

bprimem(H

140 x12 + H


120 x12 + x22)

=aprime2


h0(xx)










dx

dt= iA2x










dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































dx

dt= iA2x x(0) = x0 (64)





x isin C(RK2)int t

s


int t

s





A1 = A1 =

(V I

H0 V



minus120 is compact













j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References
































































j=0






+ the subspace

L1+ = L0










) z isin ρ(A1)



+ Hence


+ z isin ρ(A1)


+ and so






L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































L+ =(

x

Kx

) x isin D(K)

(73)



L+ =(

x

H120 x

) x isin H



ν = min

0 infx=0

(V x x)(x x)

0 (74)






equation (114)



v(τ) = T (τ)v0 τ 0 (76)











w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References




































































w(τ) = (K + V minus β)v(τ) τ 0 (710)


w(τ)Kw(τ)

)= (A1 minus β)

(v(τ)v(τ)

) τ 0


(K + V minus β)v(τ) = w(τ) (711)

(H0 + (V minus β)K)v(τ) = Kw(τ) (712)



or equivalently







and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































and (V I

H0 V

) (x0

Kx0

)= λ0

(x0

Kx0

)


















Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































Hα = W 2α2 (Rn) α isin [0 1]











V (x) =γ












2(Rn))











nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References





































































nablak =(

partk


n



n is given by

(Dpsu)(x) =( int

Rn


)1p








upk =( ksum

j=0

nablaju2p

)12

u isin W kp (Rn)



0 (Rn) u|e 1





V epp =

inte




and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References






























































and


esubRn compact

diam e1

V ep

cap(e Wmp (Rn))1p


if

limδrarr0

supesubR

n compactdiam eδ

V ep

cap(e Wmp (Rn))1p

= 0










D(A2) =(

x

y

)isin W

122 (Rn) oplus W




minus122 (Rn)

A2

(x

y

)=

(V x + y


)


122 (Rn) oplus W




minus122 (Rn)





1 minus(

a

m+ b

))m




122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References































































122 (Rn) oplus W

minus122 (Rn)


dx

dt= iA2x x(0) = x0



eitA2x0 t isin R



part

parttminus ieq

)2

minus ∆ + m2)




2 (Rn) ψ1 isinW




minus122 (Rn))



minus ipart

parttminus V



x0 =(

x(0)y(0)

)=

⎛⎝ ψ(middot 0)

minusipartψ


⎞⎠ =

(ψ0


)



122 (Rn) oplus W


dx

dt= i(V x + y) in W

122 (Rn)

dy


minus122 (Rn)




minus122 (Rn)



122 (Rn) oplus



















minus122 (Rn)


References












































































minus122 (Rn)


References





























































boris.unibe.ch · proceedings of the edinburgh mathematical society (2008) 51, 711–750 c...

Documents