Spectral perturbation bounds for selfadjoint operators I ∗ 8 Kreˇsimir Veseli´c† 0 0 2 n a Abstract J We give general spectral and eigenvalue perturbation bounds for a selfadjoint op- 1 erator perturbed in the sense of the pseudo-Friedrichs extension. We also give sev- 2 eral generalisations of the aforementioned extension. The spectral bounds for finite eigenvaluesareobtainedbyusinganalyticityandmonotonicityproperties(ratherthan ] P variational principles) and they are general enough to include eigenvalues in gaps of S theessential spectrum. . h t 1 Introduction a m The main purpose of this paper is to derive spectral and eigenvalue bounds for selfadjoint [ operators. If a selfadjoint operator H in a Hilbert space is perturbed into 2 H v T =H +A (1) T=H+A 2 7 with, say, a bounded A then the well-known spectral spectral inclusion holds 8 0 . σ(T) λ: dist(λ,σ(H)) A . (2) inclusion 4 ⊆{ ≤k k} 0 Here σ denotes the spectrum of a linear operator. (Whenever not otherwise stated we shall 7 follow the notation and the terminology of [3].) 0 : If H, A, T are finite Hermitian matrices then (1) implies v i X µ λ A , (3) ev_bound_normA k k | − |≤k k r a where µ ,λ are the non-increasingly ordered eigenvalues of T,H, respectively. (Here and k k henceforth we count the eigenvalues together with their multiplicities.) Whereas (2) may be called an upper semicontinuity bound the estimate (3) contains an existence statement: each of the intervals [λ A , λ + A ] contains ’its own’ µ . k k k −k k k k Colloquially, bounds like (2) may be called ’one-sided’ and those like (3) ’two-sided’. As it is well-known (3) can be refined to another two-sided bound minσ(A) µ λ maxσ(A). (4) ev_bound_A+- k k ≤ − ≤ In [9] the following ’relative’ two-sided bound was derived µ λ bλ , (5) vessla1_bound k k k | − |≤ | | ∗Thisworkwaspartlydoneduringtheauthor’sstayattheUniversityofSplit,FacultyofElectrotechnical Engineering,MechanicalEngineeringandNavalArchtecturewhilesupportedbytheNationalFoundationfor Science,HigherEducationandTechnologicalDevelopmentoftheRepublicofCroatia. BoththeFoundation supportandthekindhospitalityofprofessorSlapniˇcararegratefullyacknowledged. †Fernuniversita¨t Hagen, Fakulta¨t fu¨r Mathematik und Informatik Postfach 940, D-58084 Hagen, Ger- many,e-mail: [email protected]. 1 provided that (Aψ,ψ) b(H ψ,ψ), b<1. | |≤ | | This bound was found to be relevant for numerical computations. Combining (3) and (5) we obtain µ λ a+bλ , (6) ev_bound2 k k k | − |≤ | | or, equivalently, λ a bλ µ λ +a+bλ , (7) ev_bound k k k k k − − | |≤ ≤ | | provided that (Aψ,ψ) a ψ 2+b(H ψ,ψ), b<1. (8) katobound0 | |≤ k k | | One of our goals is to extend the bound (6) to general selfadjoint operators. Since these may be unbounded we have to make precise what we mean by the sum (1). Now, the condition (8) is exactly the one which guarantees the existence and the uniqueness of a closed extension T of H +A, if, say, (A) (H 1/2). The operator T is called the D ⊇ D | | pseudo-Friedrichs extension of H +A, see [3], Ch. VI. Th. 3.11. Further generalisations of this construction are containedin [2, 6, 5]. All they allowA to be merely a quadratic form, so(1)isunderstoodastheformsum;notethattheestimate(8)concernsjustforms. Partic- ularlystrikingbyitssimplicityisthe constructionmadein[5]forthe so-calledquasidefinite operators(finite matrices with this propertyhave been studied in [8], cf. also the references given there). Let H,A be bounded and, in the intuitive matrix notation, H 0 0 B H = + , A= , (9) quasidef 0 H B∗ 0 − (cid:20) − (cid:21) (cid:20) (cid:21) with H positive definite. Then ± 1 0 H 0 1 H−1B T = + + (10) schur B∗H−1 1 0 H B∗H−1B 0 1 (cid:20) + (cid:21)(cid:20) − −− + (cid:21)(cid:20) (cid:21) with an obvious bounded inverse. This is immediately transferable to unbounded H,A −1/2 −1/2 provided that F =H BH is bounded. Indeed, then (10) can be rewritten as + − 1 0 1 0 1 F T = H 1/2 H 1/2 (11) schur_unb | | F∗ 1 0 1 F∗F 0 1 | | (cid:20) (cid:21)(cid:20) − − (cid:21)(cid:20) (cid:21) which is selfadjoint as a product of factors which have bounded inverses. Note that in (8) we have a=0 and b= F and the latter need not be less than one! k k Infact, ourfirsttask will be to derive further constructionsofoperatorsdefined asform sums. One of them takes in (9) A B A= + , B∗ A − (cid:20) (cid:21) whereA areH -boundedasin(8). So,werequireb<1onlyfor’diagonalblocks’. Another ± ± one exhibits ’off-diagonal dominance’ inasmuch as H in (9) are a sort of B-bounded. All ± these constructions as well as those from [3, 2, 6, 5] are shown to be contained in a general abstract theorem which also helps to get a unified view of the material scattered in the literature. This is done in Sect. 2. Asa ruleeachsuchconstructionwillalsocontainaspectralinclusionlike(2). InSect. 3 we will give some more inclusion theorems under the condition (8) as an immediate prepa- ration for eigenvalue estimates. In the proofs the quasidefinite structure will be repeatedly used. Moreover, the decomposition (10) and the corresponding invertibility property will 2 be carried over to the Calkin algebra, thus allowing tight control of the spectral movement including the monotonicity in gaps both for the total and the essential spectra. In Sect. 4 we consider two-sided bounds for finite eigenvalues. They are obtained by using analyticity and monotonicity properties.1 In order to do this we must be able (i) to countthe eigenvalues(note thatwe maybe in agapofthe essentialspectrum)and (ii) to keep the essential spectrum away from the considered region. The condition (i) is achieved by requiring that at least one end of the considered interval be free from spectrum during the perturbation (we speak od ’impenetrability’). This will be guaranteed by one of the spectral inclusion theorems mentioned above. Similarly, (ii) is guaranteedbyanalogousinclusionsfortheessentialspectrum. Basedonthiswefirstprovea monotonicityresultforageneralclassofselfadjointholomorphicfamiliesandthenestablish the bound (6) as well as an analogous relative bound generalising (4) which includes the monotonicity of eigenvalues in spectral gaps. Another result, perhaps even more important in practice, is the one in which the form A is perturbed into B with B A small with − respectto A(this correspondstorelativelysmallperturbationsofthepotentialinquantum mechanical applications). In this case the necessary impenetrability is obtained by a con- tinuationargumentwhichassumestheknowledgeofthe wholefamilyH+ηAinsteadofthe mere unperturbed operator H +A. All our eigenvalue bounds are sharp. The corresponding eigenvector bounds as well as systematic study of applications to various particular classes of operators will be treated in forthcoming papers. Acknowledgements. The author is indebted to V. Enss, L. Grubiˇsi´c, R. Hryniv, W. Kirsch, V. Kostrykin, I. Slapniˇcar and I. Veseli´c for their helpful discusions. He is also indebted to an anonymous referee whose comments have greatly helped in the preparation of the final version of this paper. 2 Construction of operators Here we will givevarious constructionsof selfadjointoperatorsby means offorms (cf. [3,2, 6, 5]). Sometimes our results will generalise the aforementioned ones only slightly, but we will still give the proofs because their ingredients will be used in the later work. We shall include non-symmetric perturbations whenever the proofs naturally allow such possibility. gaps Definition 2.1 We say that the open interval (λ−,λ+) is a spectral gap of a selfadjoint operator H, if this interval belongs to the resolvent set ρ(H) and its ends, if finite, belong to the spectrum σ(H). The essential spectral gap is defined analogously. representation Definition 2.2 We say that a sesquilinear form τ, defined in a Hilbert space on a dense H domain represents an operator T, if D T is closed and densely defined, (12) represent1 (T), (T∗) (13) represent2 D D ⊆D (Tψ,φ)=τ(ψ,φ), ψ (T), φ , (14) represent3 ∈D ∈D (ψ,T∗φ)=τ(ψ,φ), ψ , φ (T∗). (15) represent4 ∈D ∈D uniqueness Proposition 2.3 A closed, densely defined operator T is uniquely defined by (12) – (15). 1 Another possible approach to the monotonicity could be to use variational principles valid also in spectralgaps,seee.g.[4]or[1]butwefoundtheanalyticitymoreelegant. 3 Proof. Suppose that T satisfies (12) – (15). Then 1 (Tψ,φ)=τ(ψ,φ)=(ψ,T∗φ), ψ (T), φ (T∗), 1 ∈D ∈D 1 (T ψ,φ)=τ(ψ,φ)=(ψ,T∗φ), ψ (T ), φ (T∗). 1 1 ∈D ∈D The first relation implies T T and the second T T . Q.E.D. 1 1 ⊇ ⊇ Let H be selfadjoint in a Hilbert space and let α(, ) be a sesquilinear form defined H · · on such that D α(ψ,φ) H1/2ψ H1/2φ ψ,φ (16) alphabound | |≤k 1 kk 1 k ∈D where is a core for H 1/2 and D | | H =a+bH , a,b real, b 0, H positive definite. (17) H1 1 1 | | ≥ Then the formula (Cψ,φ)=α(H−1/2ψ,H−1/2φ), ψ,φ , (18) C 1 1 ∈D defines a C ( ) with ∈B H C 1 (19) Cnorm k k≤ (note that H1/2 is dense in ). The form α can obviously be extended to the form α , 1 D H Q defined on the subspace = (H 1/2)= (H1/2) (20) Q Q D | | D 1 by the formula αQ(ψ,φ)= lim α(ψn,φn) (21) alphaQ n,m→∞ for any sequence ψ ψ, φ φ, H1/2ψ H1/2ψ, H1/2φ H1/2φ. Then (16) holds n → m → 1 n → 1 1 m → 1 for α on and Q Q (Cψ,φ)=αQ(H1−1/2ψ,H1−1/2φ), ψ,φ∈H. (22) CC The sesquilinear form for H is defined on as Q h(ψ,φ)=(J H 1/2ψ, H 1/2φ) (23) formh | | | | with J =signH. (24) Jsign In general there may be several different sign functions J of H with J2 = 1. The form h does not depend on the choice of J. pseudoF Theorem 2.4 Let H, α, C, be as above and such that D Q C =(H ζ)H−1+C (25) Czeta ζ − 1 is invertible in ( ) for some ζ C. Then the form B H ∈ τ =h+αQ (26) formtau represents a unique closed densely defined operator T whose domain is a core for H 1/2 and | | which is given by T ζ =H1/2C H1/2, (27) T-zeta − 1 ζ 1 T∗ ζ =H1/2C∗H1/2, ζ C (28) T*-zeta − 1 ζ 1 ∈ 4 and, whenever C−1 ( ), ζ ∈B H (T ζ)−1 =H−1/2C−1H−1/2 ( ), (29) invT-zeta − 1 ζ 1 ∈B H (T∗ ζ)−1 =H−1/2C−∗H−1/2 ( ). (30) invT*-zeta − 1 ζ 1 ∈B H We call T the form sum of H and α and write T =H +α. (31) THalpha If α is symmetric then T is selfadjoint. Proof. In view of what was said above we may obviously suppose that is already equal D to .2 We first prove that (H1/2C H1/2) is independent of ζ and is dense in . Indeed, Q D 1 ζ 1 H for ζ,ζ′ C and ψ (H1/2C H1/2) we have ψ and ∈ ∈D 1 ζ 1 ∈Q C H1/2ψ =(H ζ)H−1H1/2ψ+CH1/2ψ Q∋ ζ 1 − 1 1 1 =(H ζ′)H−1/2ψ+CH1/2ψ+(ζ′ ζ)H−1/2ψ − 1 1 − 1 =Cζ′H11/2ψ+(ζ′−ζ)H1−1/2ψ. Thus, by (ζ′−ζ)H1−1/2ψ ∈ Q we have Cζ′H11/2ψ ∈ Q, hence ψ ∈ D(H11/2Cζ′H11/2). Since ζ,ζ′ are arbitrary (H1/2C H1/2) is indeed independent of ζ and (27) holds. Now take ζ with C−1 ( ).DTh1en thζe 1three factors on the right hand side of (27) have bounded, ζ ∈ B H everywheredefinedinverses,so (29) holds as wellandT is closed. We now provethat (T) D is a core for H 1/2 or, equivalently, for H1/2. That is, H1/2 (T) must be dense in (see [3] III, Exerc|ise|51.9). By taking ζ with C1−1 ( ) we h1avDe H ζ ∈B H H1/2 (T)=H1/2 (T ζ)= 1 D 1 D − H1/2 ψ : C H1/2ψ =C−1 1 ∈Q ζ 1 ∈Q ζ Q n o andthisisdensebecauseC mapsbicontinuously ontoitself. Inparticular, (T)isdense ζ H D in . By H C∗ =(H ζ)H−1+C∗ ζ − 1 all properties derived above are seen to hold for T∗ as well. The identities (14), (15) follow immediately from (27) by using the obvious identity τ(ψ,φ) ζ(ψ,φ)=(C H1/2ψ,H1/2φ), (32) formtauC − ζ 1 1 valid for any ψ,φ , ζ C. Finally, if α is symmetric then T,T∗ is also symmetric and ∈ Q ∈ therefore selfadjoint. Q.E.D. pseudoFC Corollary 2.5 Let H, α, τ, T be as in Theorem 2.4. Then τ(ψ,φ)=ζ(ψ,φ) for some ψ , ζ C and all φ is equivalent to ∈Q ∈ ∈Q ψ (T), Tψ=ζψ. ∈D 2Thisassumptionwillbemadethroughout therestofthepaper,ifnotstated otherwise. 5 bounded_difference Remark 2.6 Although fairly general, the preceding theorem does not cover all relevant formrepresentations. IfT =H+αandα isanyboundedform,thenτ =τ+α obviously 1 1 1 generatesaT inthesenseofDef.2.2—weagainwriteT =H+α+α —while H, α+α 1 1 1 need not satisfy the conditions of Theorem 2.4. diag_form_bound Remark 2.7 If α is symmetric then (16) is equivalent to α(ψ,ψ) H ψ 2. (33) alphaboundsymm 1 | |≤k k In general, (33) implies (16) but with b replaced by 2b. indep_of_ab Remark 2.8 By Proposition2.3the operatorT =H+αdoes notdepend onthe choiceof a,b in the operator H from (17). Moreover, in the construction (27) H may be replaced 1 1 by any selfadjoint H =f(H) where f is a real positive-valued function and 2 a+bλ 0<m | | M < . ≤ f(λ) ≤ ∞ Then (T ζ)−1 =H−1/2D−1H−1/2, (34) invT-zeta_2 − 2 ζ 2 where D =(H ζ)H−1+D, ζ − 2 D =H1/2f(H)−1/2CH1/2f(H)−1/2 1 1 and D =H1/2f(H)−1/2C H1/2f(H)−1/2 ζ 1 ζ 1 is invertible in ( ), if and only if C is such. ζ B H pseudoF_nenciu Corollary 2.9 Let H, H1 =a+ H , , C α=αQ, h and J be as in (20) – (24) such that | | Q J +C is invertible in ( ). Then the form τ = h+α represents a unique closed densely B H defined operator T =H+αin thesenseof Remark 2.6. Moreover, (T)is acore for H 1/2 D | | and T +aJ =H1/2(J +C)H1/2 (35) Tnenciu 1 1 (and similarly for T∗). Note that the preceding construction — in contrastto the related one in Theorem 2.4 does not give an immediate representationof the resolvent, except, if a=0. In the following theorem we will use the well known formulae σ(AB) 0 =σ(BA) 0 , (36) AB_BA \{ } \{ } 1 B(λ AB)−1A (λ BA)−1 = + − (37) BA_res_AB − λ λ Bf(AB)=f(BA)B, (38) BfAB_fBAB where A,B ( ) and f is analytic on σ(AB) 0 . ∈B H ∪{ } 6 pseudoF_nenciu_factor Theorem 2.10 Let H, α, , C satisfy (16) – (22). Let, in addition, Q C =Z∗Z , Z ( ). (39) Z1Z2 2 1 1,2 ∈B H Then C from (25) is invertible in ( ), if and only if ζ B H F =1+Z H (H ζ)−1Z∗. (40) Fzeta ζ 1 1 − 2 is such. In this case Theorem 2.4 holds and (T ζ)−1 =(H ζ)−1 H1/2(H ζ)−1Z∗F−1Z H1/2(H ζ)−1. (41) invT-zeta_Z1Z2 − − − 1 − 2 ζ 1 1 − Proof. C is invertible in ( ), if and only if ζ B H 1+H (H ζ)−1C =1+H (H ζ)−1Z∗Z 1 − 1 − 2 1 is invertible in ( ). Now, B H σ(H (H ζ)−1Z∗Z ) 0 =σ(Z H (H t)−1Z∗) 0 1 − 2 1 \{ } 1 1 − 2 \{ } Hence F is invertible in ( ) if an only if C is such. In this case (29) gives ζ ζ B H (T ζ)−1 =H−1/2(1+H (H ζ)−1Z∗Z )−1H1/2(H ζ)−1 = − 1 1 − 2 1 1 − H−1/2 1 H (H ζ)−1Z∗Z (1+H (H ζ)−1Z∗Z )−1 H1/2(H ζ)−1 = 1 − 1 − 2 1 1 − 2 1 1 − (cid:0) (H ζ)−1 H1/2(H ζ)−1Z∗F−1Z H1/2(H(cid:1) ζ)−1. − − 1 − 2 ζ 1 1 − Q.E.D. We now apply Theorem 2.4 to further cases in which the key operator C from (25) is ζ invertible in ( ). B H pseudoF1 Theorem 2.11 Let H be selfadjoint and let α satisfy (16) with b < 1 and (17). Then the conditions of Theorem 2.4 are satisfied and ζ =λ+iη ρ(T) whenever ∈ a+ λb. η > | | (42) zetaregion | | √1 b2 − Proof. To prove C−1 = (H ζ)H−1+C −1 ( ) it is enough to find a ζ = λ+iη ζ − 1 ∈ B H such that (cid:0) H (H (cid:1)ζ)−1 <1. (43) etaneumann 1 k − k Now, bξ +a (H ζ)−1H supψ(ξ,a,b,λ,η), ψ(ξ,a,b,λ,η)= | | 1 k − k≤ξ∈R (ξ λ)2+η2 − A straightforward,if a bit tedious, calculation (see Appendix) shopws 1 maxψ = (a+ λb)2+b2η2 (44) maxf ξ η | | | | p Hence (42) implies (43). Q.E.D. Another similar criterion for the validity of Theorem 2.4 — oft independent of that of Theorem 2.11 is given by the following 7 V1V2cor Corollary 2.12 Let H, α, C satisfy (16)–(18) and let Z H (H ζ)−1Z∗ <1. (45) Z1Z2<1 k 1 1 − 2k for some ζ ρ(H) and with Z from (39). Then Theorem 2.10 applies. 1,2 ∈ Typically we will have α(ψ,φ)=(V ψ,V φ), (46) V1V2 1 2 where V are linear operators defined on such that 1,2 Q 1/2 Z =V H ( ). (47) Z1Z2H1 1,2 1,2 1 ∈B H In this case the formula (45) can be given a more familiar, if not always rigorous, form (cf. [6]) V (H ζ)−1V∗ <1. k 1 − 2k mygeneralisations Remark 2.13 If α(ψ,φ) = (Aψ,φ), where A is a linear operator defined on (H), D ⊆ D a core for H 1/2 then Theorem 2.11 applies and, by construction, the obtained operator D | | coincides with the one in [3] VI. Th. 3.11. The uniqueness of T as an extension of H +A, provedin [3] makes no sense in our more general,situation. Our notion of form uniqueness (which was used by [6] in the symmetric case) will be appropriate in applications to both Quantum and Continuum Mechanics. Thus, our Theorem 2.11 can be seen as a slight generalisation of [3]. On the other side, our proof of Theorem 2.4 closely follows the one from [3]. Cor. 2.9 and Th. 2.10 are essentially Theorems. 2.1, 2.2 in [6] except for the following: (i) our α need not be symmetric, (ii) we use a more general factorisability (39) instead of (46) which is supposed in [6] and finally, (iii) we need no relative compactness argument to establish Theorem 2.10. The fact that the mentioned results from [6] are covered by our theory will facilitate to handle perturbations of the form α which are not easily accessible, if α is factorised as in (46). The spectral inclusion formula (42) seems to be new. Thus, our Theorem 2.4 seems to cover essentially all known constructions thus far.3 Next we give some results on the invariance of the essential spectrum. relcomp Theorem 2.14 Let H, h, α, C, satisfy (16) – (24) with α symmetric. D Q (i) If the operator C is compact then Theorem 2.11 holds and σ (T)=σ (H). (ii) If ess ess Theorem 2.4 holds and H−1C is compact then again σ (T)=σ (H). 1 ess ess Proof. In any of the cases (i), (ii) we can find a ζ for which C−1 ( ) (in the case (i) ζ ∈ B H thisfollowsfromtheknownargumentthatforacompactC theestimate(16)willholdwith arbitrarily small b) so Theorem 2.4 holds anyway. By (29) we have (T ζ)−1 (H ζ)−1 = − − − H−1/2 ((H ζ)H−1+C)−1 H (H ζ)−1 H−1/2 = 1 − 1 − 1 − 1 (cid:0)H−1/2 (1+A)−1 1 H1/2(H ζ)−(cid:1)1 1 − 1 − where H−1A=(H ζ)−1C is co(cid:0)mpact and by C(cid:1)−1 ( ) also (1+A)−1 ( ). Hence 1 − ζ ∈B H ∈B H (T ζ)−1 (H ζ)−1 = H−1/2A(1+A)−1H1/2(H ζ)−1 − − − − 1 1 − 3Therearetwoobviousextensions: (i)addingaboundedform(Remark2.6)and(ii)multiplyingT bya bicontinuous operator. AnexampleofthelatterisT =H+αdescribedinCor.2.9. 8 is compact and the Weyl theorem applies. Q.E.D. Finally we borrow from [6] the following result which will be of interest for Dirac oper- ators with strong Coulomb potentials. relcomp_nenciu_factor Theorem 2.15 Let H, α, , C, V , Z , T be as in Theorem 2.10. Let, in addition, C 1,2 1,2 ζ Q from Theorem 2.4 be invertible in ( ) and B H 1. H have a bounded inverse, 2. the operator Z∗(H ζ)−1Z be compact for some (and then all) ζ ρ(H). 2 − 1 ∈ Then σ (T) σ (H). ess ess ⊆ The key invertibility of the operator C can be achieved in replacing the requirement ζ b<1 in (16) by some condition onthe structure of the perturbation. One such structure is given, at least symbolically, by the matrix W B + , (48) matrix B W − (cid:20) − (cid:21) where W are accretive. Such operator matrices appear in various applications (Stokes ± operator, Dirac operator, especially on a manifold ([5], [10]) and the like). Even more generalcasescould be of interest, namely those where b<1 in (16) is requiredto hold only on the “diagonal blocks” of the perturbation α. We have diagonalb Theorem 2.16 Let H, α = αQ, C, h satisfy (16) – (24) such that H has a spectral gap (λ ,λ ) containing zero. Suppose − + α(ψ,ψ) a± ψ 2+b± H d1/2ψ 2, ψ P± , (49) alpha+-0 ±ℜ ≤ k k k| − | k ∈ Q a± >0, 0<b± <1, (50) ab+- α(ψ,φ)=α(φ,ψ), ψ P+ ,φ P− . (51) alphacorner ∈ Q ∈ Q where P =(1 J)/2. Finally, suppose ± ± λ− =λ−+b− λ− <λ+ =λ+ b+ λ+ . (52) lhatless | | − | | Then τ =h+α generatesba closed, densely defibned operator T with (T) a core for H 1/2 D | | and (λ−,λ+)+iR ρ(T). (53) inclusion2 ⊆ The operator T is selfadjoint, if α is symmetric. b b Proof. We split the perturbation α into two parts α=χ+χ′ where χ is the ’symmetric diagonal part’ of α, that is, α (ψ,φ)=α(P ψ,P φ)+α(P ψ,P φ), d + + − − 1 χ(ψ,φ)= α (ψ,φ)+α (φ,ψ) . d d 2 (cid:16) (cid:17) 9 Symbolically,4 χ 0 h 0 χ= + , h= + . 0 χ 0 h − − (cid:20) − (cid:21) (cid:20) − (cid:21) Now (49) and the standard perturbation result for closed symmetric forms ([3] Ch. VI, Th. 3.6) implies h =h +χ is symmetric, bounded from below by ± ± ± e λ± b± λ± a± ± − | |− andclosedon . ThethusgeneratedselfadjointoperatorH has (H 1/2)=P . Now, ± ± ± Q D | | Q h e 0 e τ =h+α=h+χ′, h= + . " 0 h− # − e e e We write e h 0 H 0 τ =h+α=h+χ′, h= + , H = + , " 0 h− # " 0 H− # − − e e e e e where H has a spectral gap contained in (λ ,λe) and e − + e J =sign(H d)=esigenH, λ− <d<λ+. − WewillapplyTheorem2.4toH,χe′. WehavefirsttoeprovethateH,χ′ satisfytheconditions (16), (17) (possibly with different constants a,b). By (49) we have e e a h ± ± 0 h + . ± ≤ ≤ 1 b 1 b ± ± − − e Hence H cH d | |≤ | − | for any d (λ ,λ ) and some c = c(d). So, H,χ′ satisfy (16), (17) with H replaced by ∈ − + e | | H d. We take ζ =d+iη and set | − | e e e e T ζ = H d1/2Dζ H d1/2 (54) Ttilde − | − | | − | e Dζ =Je ζ H d−1e+D, − | − | (Dψ,φ)=χ′(H d−1/2ψ, H d−1/2φ), | − |e | − | D F D =e + e . F∗ D − (cid:20) − (cid:21) By the construction we have χ′(P±ψ,P±ψ)=0. (55) +-chi’accretive ℜ Hence D are skew Hermitian and ± 1 iη(H d)−1+D F Dζ =" − +− F+∗ 1 i(d H−)−1 D− # − − − − e 4Throughoutthispaperwewillfreelyusematrixnotationforboundedeoperatorsaswellasforunbounded ones orformswhenever thelatter areunambigouslydefined. Thematrixpartitionreferstotheorthogonal decompositionH=P+H⊕P−H. 10