PT-Symmetric Schrödinger Operators with Unbounded Potentials

Jan Nesemann PT-Symmetric Schrödinger Operators with Unbounded Potentials VIEWEG+TEUBNER RESEARCH Jan Nesemann PT-...

Author: Jan Nesemann

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Jan Nesemann PT-Symmetric Schrödinger Operators with Unbounded Potentials

VIEWEG+TEUBNER RESEARCH

Jan Nesemann

PT-Symmetric Schrödinger Operators with Unbounded Potentials

VIEWEG+TEUBNER RESEARCH

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

Dissertation Universität Bern, 2010

1st Edition 2011 All rights reserved © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011 Editorial Office: Ute Wrasmann |Anita Wilke Vieweg+Teubner Verlag is a brand of Springer Fachmedien. Springer Fachmedien is part of Springer Science+Business Media. www.viewegteubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printed on acid-free paper Printed in Germany ISBN 978-3-8348-1762-4

Acknowledgment I would like to express my gratitude to my supervisor Prof. Dr. C. Tretter for giving me the opportunity to write this thesis and for helping and supporting me during the past years. In particular, I would like to thank her for providing my wife and me with the great chance to come here to Switzerland. This was a welcome change which put us in the position to strike new paths. Also, I would like to thank the members of the Applied Analysis Group for their help. I am deeply grateful to my wife, Kerstin, for her encouragement and love. The work on this thesis was financially supported by the German Research Foundation (DFG), grant number TR368/6-1, and the Swiss National Science Foundation (SNSF), grant number 200021-119826/1. Jan Nesemann

Table of Contents Acknowledgment

V

Introduction

1

1 Relatively Bounded Perturbations in Krein Spaces 1.1 Linear Operators in Krein Spaces . . . . . . . . . . . . . . . . . . 1.2 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Relatively Bounded and Relatively Compact Operators 1.2.2 The Case of Relative Bound 0 . . . . . . . . . . . . . . . . 1.2.3 Stability of Self-Adjointness in Krein Spaces . . . . . . . 1.3 Continuity of Separated Parts of the Spectrum . . . . . . . . . . 1.3.1 Continuity of Resolvents . . . . . . . . . . . . . . . . . . . 1.3.2 Perturbation of Isolated Parts of the Spectrum . . . . . . 1.3.3 Perturbation of Spectra of Self-Adjoint Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 14 14 17 19 20 20 23

2 Relatively Form-Bounded Perturbations in Krein Spaces 2.1 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Accretive and Sectorial Operators . . . . . . . . . . . . . 2.1.2 Quadratic Forms and Associated Operators . . . . . . . . 2.1.3 Relatively Form-Bounded and Relatively Form-Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Continuity of Separated Parts of the Spectrum . . . . . . . . . . 2.2.1 Perturbation of Spectra of Self-Adjoint Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Pseudo-Friedrichs Extensions . . . . . . . . . . . . . . . . . . . . 2.3.1 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 35 36

26 30

42 46 47 53 59 59

VIII

Table of Contents

3 Examples 3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Class of Schrödinger Operators with Relatively Bounded Complex Potentials and Real Spectrum . . . . . . . . . . . . . .

65 65 69

Bibliography

79

72

Introduction In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H ∗ . The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H, is real but it is not a necessary condition. In the literature on so-called PT-symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs. From a mathematical point of view, however, this is no surprise at all – provided one is familiar with the theory of self-adjoint operators in spaces with indefinite inner product (Krein spaces). The physical structure found to be the reason for the reality of the spectrum is PT-symmetry (spacetime reflection symmetry), which amounts to self-adjointness in some Krein space. A Hamiltonian H is PT-symmetric if it commutes with PT, that is PT H = HPT, compare, e.g., [Ben07] and [AT10]. Here P denotes the space reflection (parity) operator and T the time reflection operator. P and T satisfy the relations P 2 = T 2 = (PT)2 = I and PT = TP. If p = id/dx and x are the momentum and position operators, then P has the effect p → − p,

x → − x

and T has the effect p → − p,

x → x,

i → −i,

compare, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]. In contrast to self-adjointness in Hilbert spaces, PT-symmetry does not necessarily lead to a completely real spectrum. For example, the Hamiltonian H := p2 + ix3 J. Nesemann, PT-Symmetric Schrödinger Operators with Unbounded Potentials, DOI 10.1007/978-3-8348-8327-8_1, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

2

Introduction

is not symmetric in the Hilbert space L2 (R) since the potential is not real-valued. However, the Hamiltonian H is PT-symmetric in the Hilbert space L2 (R): PT H(PT)−1 f (x) = PT HTP f (x) = PT HT f (− x) = PT H f (− x) = PT p2 f (− x) + ix3 f (− x) = P p2 f (− x) − ix3 f (− x) = p 2 f (x) − i(− x)3 f (x) = H f (x),

f ∈ D (H),

where D (H) is the maximal domain of H. More generally, for the family of PT-symmetric Hamiltonians (compare, e.g., [BB98] and [AT10]) Hε := p2 + x2 (ix)ε ,

ε ∈ R,

the spectrum of Hε was found to be real and positive if ε ≥ 0 and partly real and partly complex if ε < 0 (see, e.g., [DDT01a] for a proof of the reality of the spectrum for ε ≥ 0; for ε < 0 numerical results indicate the appearance of complex eigenvalues, see, e.g., [BB98] and [Ben04a]). More precisely, for −1 < ε < 0, there is a finite number of real positive eigenvalues and an infinite number of complex conjugate pairs of eigenvalues, if ε ≤ −1, then there are no real eigenvalues, see, e.g., [BB98] and [Ben04a]. During the last decade PT-symmetric models have been analysed intensively, see, e.g., the review paper [Ben07] and the references therein. Within the vast literature on PT-symmetric problems there are only some mathematically rigorous papers, see, e.g. [DDT01b], [Shi02], [AK04], [LT04], [Shi04], [Shi05], [Tan06], [Tan07] and [AT10]. In particular, we mention the works of E. Caliceti, F. Cannata, S. Graffi and J. Sjöstrand (see [Cal04], [CGS05], [CG05], [Cal05], [CCG06], [CG08] and [CCG08]), who use perturbation theory for linear operators. In [Mos02], [Jap02], [AK04], [LT04], [GSZ05] and [Tan06] Krein space methods were applied to PT-symmetric problems. The paper by H. Langer and C. Tretter (see [LT04] and [LT06]) was the first where Krein space methods were used to prove rigorous abstract results for PT-symmetric problems; this approach is also crucial for this thesis. Consider the following situation. If a self-adjoint operator A 0 in a Krein space K , [·, ·] , which is also self-adjoint with respect to some Hilbert space inner product (·, ·) on K , has an isolated real eigenvalue of definite type,

Introduction

3

then this eigenvalue remains real under a “sufficiently small” PT-symmetric perturbation V (that is, V is symmetric in K ,[·, ·] ). This theorem has been proven for the case of bounded V in [LT04] and relies on the fact that a uniformly positive subspace of a Krein space is stable, which is a well-known result in the theory of Krein spaces. The theorem mentioned above can be applied to isolated eigenvalues of PT-symmetric problems. If two simple real eigenvalues of the same type meet, they remain real after crossing. This is the case of self-adjoint operators in Hilbert spaces, where all eigenvalues are of positive type. If two real eigenvalues of different type meet, they will, in general, develop into a pair of non-real complex conjugate eigenvalues. While the case of bounded V was treated in [LT04] (see also [LT06]), a comparable result for the case of unbounded V has been missing. The case of unbounded potentials has only been considered for a few special classes or examples of operators, see, e.g., [DDT01b], [Shi02], [CG05], [Cal05] and [CG08]. The aim of this thesis is to generalize the results of [LT04] to wide classes of unbounded potentials, e.g., to relatively bounded and relatively form-bounded operators. This includes a generalization of the results obtained in [CG05] for a special class of Schrödinger operators with relatively bounded complex polynomial potentials. The main results of this thesis are stability results for the reality of the spectrum of a family of operators A ε of the form A ε := A 0 + εV ,

ε ∈ [0,1];

in particular, we consider the case where A ε is self-adjoint in a Krein space K , [·, ·] while A 0 is also self-adjoint with respect to some Hilbert space inner product (·, ·) on K . Furthermore, we give inclusions for the perturbed spectrum of A ε . We found different assumptions on V to prove the respective results. More precisely, we consider the following three types of assumptions on V ; in any case V is assumed to be symmetric in the Krein space K , [·, ·] . (a) D (A 0 ) ⊂ D (V ) and there exist constants α ≥ 0, 0 ≤ β < 1/2 such that (1)

V x ≤ α x + β A 0 x ,

x ∈ D (A 0 );

in this case A ε = A 0 + V is defined as an operator sum. (b) A 0 and V are bounded from below in K , [·, ·] , D (a0 ) ⊂ D (v) for the quadratic forms a0 and v associated with A 0 and V , respectively, and there exist constants α ≥ 0, 0 ≤ β < 1/2 such that |v x| ≤ α x 2 + β|a0 x|,

x ∈ D (a0 );

4

Introduction

in this case A ε = A 0 V is defined as a form sum, which is an extension of the operator sum. (c) D (V ) ⊂ D (A 0 ) and there exist constants α ≥ 0, 0 ≤ β < 1/2 such that |[V x, x]| ≤ α x 2 + β [J | A 0 | x, x],

x ∈ D (V ),

where J denotes a fundamental symmetry on K , and D (V ) is a core of | A 0 |1/2 ; in this case A ε is the pseudo-Friedrichs extension of A 0 + εV . For example, in terms of relative boundedness properties of V with respect to A 0 , case (a), our main results are the following. Since, by assumption, A 0 is self-adjoint in a Hilbert space, its spectrum is real. We establish the following conditions which guarantee the spectrum of A 0 + V to be real, even when A 0 + V is not self-adjoint in a Hilbert space (compare Theorem 1.44 below): (i) Suppose λ0 is an isolated eigenvalue of A 0 of definite type with finite multiplicity m. If (2)

1 1 α + β δ + |λ0 | < , δ 2

where δ := dist λ0 , σ(A 0 )\{λ0 } , then σ(A 1 ) ∩ Bδ/2 (λ0 ) consists of a finite system of isolated and real eigenvalues with total multiplicity m which are of the same type as λ0 .

(ii) The preceding result can be extended to the case when the spectrum of A 0 is discrete and consists of an infinite sequence of eigenvalues · · · < λ0−2 < λ0−1 < λ01 < λ02 < · · ·

of definite type with finite multiplicities. In this case it is necessary that (2) holds for each eigenvalue λ0n , n ∈ Z∗ . Let δn := dist λ0n , σ(A 0 )\{λ0n } , n ∈ Z∗ , and suppose that (1) holds with αn ≥ 0 and βn ∈ [0, 1/2), n ∈ Z∗ , such that 1 0 γ := sup αn + βn δn + |λ n | < ∞. n∈Z∗ δn

(3)

Then the spectrum of A ε is discrete and consists of real eigenvalues which are of definite type for all ε ∈ [0, ε0 ], where ε0 ∈ (0, 1] has to be chosen such that ε0 < 1/(2γ).

Introduction

5

The preceding result can be illustrated by the following example (see Section 3.3 below) which was first studied in [CG05]. Consider operators induced by the differential expression Aε = −

d2 + P + ε iQ, dx2

ε ∈ [0, 1],

in L2 (R), where P and Q are multiplication operators by real polynomials P and Q; P is an even polynomial of degree 2p, p ≥ 1, with lim|x|→∞ P(x) = ∞, and Q is an odd polynomial of degree 2q − 1, q ≥ 1, such that p > 2q. In this special case the assumptions of (ii) are satisfied for A0 = −

d2 +P dx2

and

V = iQ;

the spectrum of A 0 consists of an infinite sequence of eigenvalues λ01 < λ02 < · · · and the constants αn ≥ 0 and βn ∈ [0,1/2), n ∈ N, in (1) can be chosen such that (3) holds. The results (i) and (ii) can be extended to the case where isolated compact parts of the spectrum of A 0 are considered instead of isolated eigenvalues (see Theorem 1.46 below). Furthermore, the results remain valid for cases (b) of relatively form-bounded operators and (c) of pseudo-Friedrichs extensions. The proof of the results (i) and (ii) relies on the fact that a uniformly positive subspace of a Krein space is stable (see [LT04, Theorem 3.1]). This stability theorem applies to isolated eigenvalues or isolated (compact) parts of the spectrum of the operator family A ε . In order to ensure that isolated eigenvalues or isolated parts of the spectrum of A 0 remain isolated under the perturbation εV , it is necessary that the perturbation εV is “sufficiently small” or, equivalently, A ε is “sufficiently close” to A 0 . While the “distance” between two bounded linear operators can be defined as the norm of their difference, the “distance” between two unbounded linear operators has to be measured in a different way. To this end the notion of generalized convergence is used, which amounts to convergence between the graphs of two unbounded linear operators or, equivalently, to the convergence of the resolvent of A ε to the resolvent of A 0 in norm. The latter is guaranteed by assuming that V is relatively bounded (or relatively form-bounded, respectively) with respect to A 0 with relative bound (relative form-bound, respectively) less than 1. The results achieved in this thesis are new in various aspects. In cases (a) and (b), results were known only for very particular classes of differential operators (compare [CG05] and [CG08], respectively). For case (c) the

6

Introduction

results of this thesis have been shown before in [Ves72a] and [Ves72b], but the proofs were different. In comparison to our results, [Ves72b] requires further assumptions but shows in addition to the reality of the spectrum of the pseudo-Friedrichs extension A 1 , A 1 is similar to a self-adjoint operator in a Hilbert space, compare Remark 2.52 below. The thesis is organized as follows. In Chapter 1 the case (a) of relatively bounded V is considered. The first section of this chapter gives a brief introduction into the theory of linear operators in Krein spaces. Subsequently, the reader is provided with fundamental definitions as well as elementary facts for relatively bounded operators. In Section 3 we introduce the notion of generalized convergence and we present a proof of the well-known result that, for an arbitrary family of closed linear operators Tε , ε ∈ [0, 1], in a Banach space, Tε converges to T0 in the generalized sense if and only if the resolvent of Tε converges to the resolvent of T0 in norm. Further, we recall important results from perturbation theory regarding the change of the spectrum. If a Cauchy contour Γ separates a bounded part of the spectrum σ(T0 ) of T0 from the rest and Tε converges to T0 in the generalized sense, the spectrum of Tε is likewise separated into two parts by Γ, moreover, the isolated part enclosed by Γ changes continuously with ε. If V is A 0 -bounded with A 0 -bound less than 1, then A ε converges to A 0 in the generalized sense and hence the above results apply to the family of operators A ε = A 0 + εV . Consequently, isolated eigenvalues or isolated parts of the spectrum of A 0 remain isolated under the perturbation εV . This enables us to apply the Krein space methods of [LT04] to establish criteria for the operator A ε to have real spectrum consisting of isolated eigenvalues or isolated parts if this holds for A 0 . Chapter 2 extends the results of Chapter 1 to the case (b) of relatively form-bounded perturbations. Instead of studying the usual operator sum A 0 + εV , we consider the sum A 0 εV of A 0 and εV defined by means of quadratic forms which is an extension of the operator sum A 0 + εV . While the condition of relative form-boundedness itself is less restrictive than the one of relative boundedness, relatively form-bounded operators have to be required to be bounded from below (with respect to the respective inner product); therefore, case (b) constitutes a different class of unbounded perturbations compared to case (a). Nevertheless, as in case (a), relative form-boundedness of V with respect to A 0 with relative form-bound less than 1 guarantees that A ε converges to A 0 in the generalized sense. This enables us to extend the results of Chapter 1 to the case of relatively form-bounded operators.

Introduction

7

At the end of the second chapter, we consider case (c) and introduce the notion of pseudo-Friedrichs extensions. A pseudo-Friedrichs extension is an extension of the usual operator sum which is different from the form-sum introduced before; in particular, the domain inclusion is D (V ) ⊂ D (A 0 ) rather than D (A 0 ) ⊂ D (V ) (case (a)) or D (a0 ) ⊂ D (v) (case (b)). The results are not essentially related to sesquilinear forms, but the techniques used in the proofs are similar. In the context of Krein spaces, these operators have also been studied in [Ves72a], [Ves72b] and [Ves08] where similar results were obtained, but by different proofs. In Chapter 3 we present some examples where the results of this thesis are applied to ordinary differential operators. In Sections 3.1 and 3.2 we study a second and a fourth order differential operator, respectively, on a compact interval. The class of differential operators on R introduced in [CG05] which is also covered by the results of this thesis is considered in Section 3.3. For all these examples the results show that the spectrum of A 0 + V remains real, even though A 0 + V is not self-adjoint in a Hilbert space. Notation. For an introduction to the theory of unbounded linear operators, the following notation and basic terminology as well as for further details we refer to [Kat95], [GGK90] and [RS80, RS75, RS79, RS78]. The domain of a linear operator A in a Banach space X we denote by D (A), the range of A by R (A) and the graph of A by G (A). If A is a closed linear operator, we denote the spectrum and the resolvent set of A by σ(A) and ρ (A), respectively.

Chapter 1 Relatively Bounded Perturbations in Krein Spaces 1.1 Linear Operators in Krein Spaces The main results of this thesis are based on the theory of linear operators acting in Krein spaces. In this section we briefly outline the definitions and some elementary facts. A detailed study of Krein spaces and linear operators therein can be found in [Lan62], [Lan82], [AI89], [Bog74] and [And79]. Definition 1.1. An inner product space K ,[·, ·] is called Krein space if it contains two subspaces H + , H − with the following properties:

˙ ]H − , K = H + [+

(1.1) (1.2)

H + ,[·, ·] and H − , −[·, ·] are Hilbert spaces;

˙ ] denotes the direct [·, ·]-orthogonal sum. here [+ Condition (1.2) means that the inner product is positive definite on H + and negative definite on H − , i.e., [x, x] > 0 for x ∈ H + , x = 0, [x, x] < 0 for x ∈ H − , x = 0, and that H + and H − are complete with respect to the norms x = [x, x]1/2 , x ∈ H + and x = (−[x, x])1/2 , x ∈ H − , respectively. According to this definition, the class of Krein spaces includes Hilbert spaces for which H − = {0}. The decomposition (1.1) is not unique and each decomposition (1.1) defines a Hilbert space inner product on K by the relation (1.3)

(x, y) := [x+ , y+ ] − [x− , y− ],

x = x+ + x− , y = y+ + y− ,

x± , y± ∈ H ± .

Although these inner products depend on the decomposition (1.1) the norms generated by them are all equivalent (see [Lan82, Proposition I.1.2]). Any of J. Nesemann, PT-Symmetric Schrödinger Operators with Unbounded Potentials, DOI 10.1007/978-3-8348-8327-8_2, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

10

Chapter 1 Relatively Bounded Perturbations in Krein Spaces

these norms is denoted by · , and so is the corresponding operator norm. If no other topology is explicitly mentioned, all the topological notions refer to this Hilbert space norm topology. For example, a subspace of K is a linear manifold in K which is closed with respect to this Hilbert space topology, continuity of operators means continuity with respect to this Hilbert space topology et cetera. Definition 1.2. Let K ,[·, ·] be a Krein space. A bounded linear operator J in K such that J x := x+ − x− if x = x+ + x− , x± ∈ H ± , is called fundamental symmetry corresponding to a decomposition (1.1). The relationship between a Krein space , [ · , · ] and the Hilbert space K K , (·, ·) arising from K using any of the equivalent Hilbert space inner products (·, ·) defined by (1.3) can be described as follows. We introduce the linear operators

P± x := x±

if

x = x+ + x− , x± ∈ H ± ,

which are the (·, ·)-orthogonal projections onto H + and H − , respectively. Then we have, for the fundamental symmetry J = P+ − P− corresponding to (1.1), (1.4)

[x, y] = [x+ , y+ ] − [− x− , y− ] = (x+ − x− , y+ + y− ) = (J x, y)

as well as (1.5)

(x, y) = [x+ , y+ ] + [− x− , y− ] = [x+ − x− , y+ + y− ] = [J x, y]

for x = x+ + x− and y = y+ + y− with x± , y± ∈ H ± . Furthermore, J 2 = P+ + P− = I and J = J ∗ = J −1 , where ∗ denotes the adjoint with respect to the Hilbert space inner product (1.3). Remark 1.3. In the following, for a given Krein space K , [·, ·] by K , (·, ·) we always denote the Hilbert space corresponding to K , [·, ·] as described above. Definition 1.4. Let K , [·, ·] be a Krein space. An element x ∈ K is called

(i) positive if [x, x] > 0, (ii) negative if [x, x] < 0 and (iii) neutral if [x, x] = 0.

1.1 Linear Operators in Krein Spaces

11

A subspace L of K is called non-negative (non-positive, respectively) if all elements of L are not negative (not positive, respectively); L is said to be positive (negative, respectively) if all the non-zero elements of L are positive (negative, respectively), and L is called neutral if [x, x] = 0 for all x ∈ L . Further, L is called uniformly positive if there exists a constant γ > 0 such that [x, x] ≥ γ x 2 , x ∈ L , and, analogously, uniformly negative if there exists a constant γ > 0 such that [x, x] ≤ −γ x 2 , x ∈ L . Here · denotes any of the equivalent Hilbert space norms generated by a decomposition (1.1). A subspace L of K is called definite if it is either positive or negative, and indefinite if it is neither positive nor negative. The term uniformly definite is defined accordingly. In order to define adjoint operators with respect to the indefinite inner product [·, ·], we need the following analogue of F. Riesz’ representation theorem, which follows from the latter using (1.4) and (1.5). Theorem 1.5 (F. Riesz’ representation theorem). Let (K , [·, ·]) be a Krein space and f a bounded linear functional on K . Then there exists a uniquely determined element y f ∈ K such that f (x) = [x, y f ],

x∈K ,

and f = y f . Conversely, every y ∈ K defines a bounded linear functional f y on K by f y (x) := [x, y],

x∈K ,

with f y = y . Proof. The proof follows from the Hilbert space version of F. Riesz’ representation theorem. If f is a bounded linear functional on K , then, since boundedness refers to the Hilbert space norm on K , there exists a unique yf ∈ K such that f (x) = (x, yf ), x ∈ K , and f = yf . Now the claim follows from (1.4) and (1.5) with yf := J yf . Conversely, let y ∈ K be arbitrary and set y := J y ∈ K . Then, by the Hilbert space version of the theorem, f y (x) := [x, y] = (x, J y) = (x, y ) is a bounded linear functional on K with f y = y = y . ■

12


Corollary 1.6. A Krein space K , [·, ·] is reflexive.

Proof. Let y ∈ K , the bidual space of K . According to Theorem 1.5 applied to K there exists a y ∈ K such that for all x ∈ K y (x ) = [x , y ]K = [y, x]K = f x (y) = x (y) = i K (y) (x ), where i K : K → K is the canonical embedding of K into its bidual space. That is, i K : K → K is surjective. ■ Definition 1.7. Let K , [·, ·] be a Krein space and A a densely defined linear operator in K . Then, the Krein space adjoint A [∗] of A is defined by D A [∗] := y ∈ K : [A ·, y] is continuous on D (A) , [Ax, y] = [x, A [∗] y], x ∈ D (A), y ∈ D A [∗] .

Further, A is called symmetric (in the Krein space K ) if A ⊂ A [∗] and self-adjoint if A = A [∗] . In the following we always denote the Krein space adjoint (compare, e.g., [Lan82]) by [∗] and the Hilbert space adjoint by ∗ . If J denotes a fundamental symmetry corresponding to (1.1), then the Krein space adjoint A [∗] is sometimes also called J-adjoint because of the identity (1.6)

A [∗] = J A ∗ J

(see [Lan62, Lemma I.5] or [Lan82, Paragraph I.3]). Correspondingly, A is also called J-symmetric if A ⊂ A [∗] and J-self-adjoint if A = A [∗] . The following lemma gives a convenient relation between self-adjointness of linear operators in Krein and Hilbert spaces. in a Krein space Lemma 1.8. Let A be a densely defined linear operator K , [·, ·] . Further let the bounded linear operator J in K , [·, ·] be a fundamental symmetry. Then A is self-adjoint (symmetric, respectively) with respect to the Krein space inner product [·, ·] if and only if J A is self-adjoint (symmetric, respectively) with respect to the Hilbert space inner product (·, ·). Proof. Since J is bounded, we have D J A = D A and (J A)∗ = A ∗ J ∗ (see, e.g., [MV97, Lemma 19.9]). If A ⊂ A [∗] , then, by (1.4), (J Ax, y) = [Ax, y] = [x, A y] = (J x, A y) = (x, J A y), x, y ∈ D J A = D A ,

1.1 Linear Operators in Krein Spaces

13

which implies that J A ⊂ (J A)∗ , i.e., J A is symmetric with respect to the Hilbert space inner product (·, ·) on K . If A = A [∗] , then, in addition, D J A = D J A [∗] = D J J A ∗ J = D A ∗ J ∗ = D (J A)∗ , i.e., J A = (J A)∗ . Vice versa, if J A ⊂ (J A)∗ , then [Ax, y] = (J Ax, y) = (x, J A y) = [J x, J A y] = [x, A y],

x, y ∈ D A ,

which implies that J A ⊂ (J A)[∗] , i.e., J A is symmetric with respect to the Krein space inner product [·, ·] on K . If J A = (J A)∗ , then, in addition, D A = D J 2 A = D J(J A)∗ = D J A ∗ J ∗ = D J A ∗ J = D A [∗] , i.e., A = A [∗] .

■

While the spectrum of a self-adjoint operator in a Hilbert space is always real, the spectrum of a self-adjoint operator in a Krein space may be complex. However, it is well-known that it is symmetric with respect to the real axis (see [Lan82, Paragraph I.3]). Since this property is crucial in the following we include its proof. Here and in the sequel, for a subset M ⊂ C, we denote by M ∗ := z : z ∈ M the set complex conjugate to M . Theorem 1.9. Let A be a self-adjoint operator in a Krein space K ,[·, ·] . Then the spectrum σ(A) of A is symmetric with respect to the real axis.

Proof. Using relation (1.6) we obtain for λ ∈ ρ (A)

(A − λ I)−1

[∗]

∗ −1 −1 = J (A − λ I)−1 J = J (A − λ I)∗ J = J(A − λ I)∗ J −1 [∗] −1 = (A − λ I)[∗] = A − λI ,

which yields σ(A [∗] ) = σ(A)∗ . Hence σ(A) = σ(A)∗ , since A is self-adjoint in ■ K , [ ·, ·] . Definition 1.10. Let K , [·, ·] be a Krein space and A be a self-adjoint operator in K . A real isolated eigenvalue λ is called of positive type (negative type, respectively) if the corresponding algebraic eigenspace Lλ is positive

14


(negative, respectively), and λ is called critical if Lλ is neutral. If an eigenvalue is either of positive or of negative type it is called of definite type. If for an eigenvalue λ the dimension of the corresponding algebraic eigenspace L λ is finite, then the multiplicity of λ is defined as the dimension of L λ . The total multiplicity of a finite set of eigenvalues is the sum of all its multiplicities or the dimension of the linear span of all the corresponding algebraic eigenspaces.

1.2 Stability Theorems In this paragraph the notion of relatively bounded and relatively compact operators in Banach spaces is recalled. A number of important properties of linear operators are preserved under relatively bounded or relatively compact perturbations. For a detailed study of relatively bounded and relatively compact operators see [Kat95], [EE87], [GGK90] and [Gol66].

1.2.1 Relatively Bounded and Relatively Compact Operators Definition 1.11. Let X , Y1 and Y2 be Banach spaces. Further, let A and V be linear operators, A from X to Y1 and V from X to Y2 . If D (A) ⊂ D (V ) and there exist non-negative constants α and β such that (1.7)

V x ≤ α x + β Ax ,

x ∈ D (A),

then V is called relatively bounded with respect to A or simply A-bounded. The greatest lower bound β0 of all possible constants β in (1.7) is called relative bound of V with respect to A or simply A-bound, i.e., β0 := inf β ≥ 0 : ∃ α ≥ 0 such that V x ≤ α x + β Ax , x ∈ D (A) .

Clearly, a bounded everywhere defined operator V is relatively bounded with respect to any operator A with relative bound 0. The converse is not true: in general, it is not possible to choose β = β0 in inequality (1.7) (compare also [Kat95, Example IV.1.6]).

1.2 Stability Theorems

15

Remark 1.12. If V is relatively bounded with respect to A with A-bound β0 ≥ 0, then for any β > β0 there exists an αβ > 0 such that V x ≤ αβ x + β Ax ,

(1.8)

x ∈ D (A).

An alternative to inequality (1.7) and to determine the relative bound is given by the following lemma (see [Kat95, Section V.4.1]). Lemma 1.13. The following statements are equivalent: (i) ∃ α, β ≥ 0 such that V x ≤ α x + β Ax , x ∈ D (A), (ii) ∃ α , β ≥ 0 such that V x 2 ≤ α2 x 2 + β2 Ax 2 , x ∈ D (A). Moreover,

β0 = inf β ≥ 0: ∃α ≥ 0 such that V x ≤ α x + β Ax , x ∈ D (A) (1.9) = inf β ≥ 0: ∃α ≥ 0 such that V x 2 ≤ α2 x 2 + β2 Ax 2 , x ∈ D (A) .

Proof. If (ii) holds, then, obviously, 2 V x 2 ≤ α2 x 2 + β2 Ax 2 ≤ α x + β Ax ,

x ∈ D (A),

and so (i) holds with α := α , β := β . Vice versa, suppose that (i) holds and let ε > 0 be arbitrary. Then we have the estimates V x 2 ≤ α2 x 2 + β2 Ax 2 + 2αβ x Ax

2 α ≤ α2 x 2 + β2 Ax 2 + 2αβ x Ax + x − εβ Ax ε

α2 = α2 x 2 + β2 Ax 2 + x 2 + εβ2 Ax 2 ε 1 2 = 1 + α x 2 + (1 + ε)β2 Ax 2 , x ∈ D (A). ε

Hence (ii) holds with α := 1 + 1ε α2 ≥ 0 and β := (1 + ε)β2 ≥ 0. Since ε > 0 is arbitrary, the A-bound of V may as well be defined as the greatest lower bound of the possible values of β . ■

Definition and Remark 1.14. Let X and Y be Banach spaces and let A be a closed linear operator from X to Y . Set x A := x + Ax ,

x ∈ D (A).

16


Then · A defines a norm on D (A) which is called graph norm (and sometimes also A-norm). Further, D (A), · A becomes a Banach space, which we denote by D A . Proof. Indeed · A defines a norm on D (A) since · is a norm and A is linear. By assumption, A is closed, i.e., given a sequence (xn )∞ n=1 ∈ D (A) such that xn → x and Axn → y, it follows that x ∈ D (A) and Ax = y. Hence each Cauchy sequence in D (A) converges in D (A), i.e., D (A), · A is a Banach space. ■ Remark 1.15. Let X , Y1 and Y2 be Banach spaces and let A be a closed linear operator from X to Y1 . Suppose that V is a linear operator from X to Y2 such that D (A) ⊂ D (V ). Then the restriction of V to D (A) can be regarded from D A to Y2 . By Definition and Remark 1.14, V is as a linear operator V bounded if and only if V is A-bounded. Proof. Let V be A-bounded and let α, β such that (1.7) holds. Then

V x = V x ≤ α x + β Ax ≤ max{α, β} x A , x ∈ D (A). (1.10) is bounded, then Vice versa, if V

x A = V x ≤ V x + V Ax , V x = V

. Hence (1.7) holds with α := β := V

x ∈ D (A). ■

Remark 1.16. If A is closed and V is closable, then D (A) ⊂ D (V ) implies that V is A-bounded. be defined as in Remark 1.15. According to Remark 1.15, V is Proof. Let V is bounded in D A . It is thus sufficient to show that A-bounded if and only if V is closed in D A , since then, by the Closed Graph Theorem, V is bounded in V · A ∞ xn → y for some y ∈ Y2 . Since D A . Let (xn ) ⊂ D (A) such that xn → 0 and V n=0

· A · xn = V xn → 0, that is, V is xn → 0 also xn → 0. Hence, since V is closable, V is closed since V is defined on all of D (A) = D A . ■ closable. Consequently, V

We note the following obvious relation. Remark 1.17. Let A and V be linear operators in a Krein space K . Then V is A-bounded with A-bound β0 if and only if JV is J A-bounded with J Abound β0 .


17

Proof. The proof follows from the identity J x = x , x ∈ K .

■

Definition 1.18. Let X , Y1 and Y2 be Banach spaces. Further, let A and V be linear operators, A from X to Y1 and V from X to Y2 . If D (A) ⊂ D (V ) and ∞ ∞ for any sequence (xn )∞ n=1 ⊂ D (A) such that (x n ) n=1 and (Ax n ) n=1 are bounded, ∞ (V xn )n=1 has a convergent subsequence, then V is called relatively compact with respect to A or simply A-compact. Remark 1.19. If V is A-compact, then V is A-bounded. Proof. Assume V is not A-bounded. Then there exists a sequence (xn )∞ n=1 ⊂ D (A) such that xn A = xn + Axn = 1 but V xn ≥ n for all n ∈ N. Since (V xn )∞ n=1 does not have a convergent subsequence, which is obvious, this is a ■ contradiction to the A-compactness of V . Remark 1.20. Let X , Y1 and Y2 be Banach spaces and let A be a closed linear operator from X to Y1 . Suppose that V is a linear operator from X to of V to D (A) defined as in Y2 such that D (A) ⊂ D (V ). Then the restriction V Remark 1.15 is compact if and only if V is A-compact. ∞ Proof. The claim is obvious since (xn )∞ 0 is bounded in D A if and only if (x n )0 and (Axn )∞ are bounded in X and Y , respectively, and V = V | . ■ 1 D (A) 0

1.2.2 The Case of Relative Bound 0 In the following we are particularly interested in relatively bounded operators having relative bound 0. Necessary conditions for this special case can be found in the literature, see [Hes69], [Wei00, Sec. 9.2], [EE87, Sec. III.7] and [Gol66, Sec. V.3]. A Hilbert space version of the following lemma and its proof can be found in [Jör67], see Hilfssatz 1.1 therein. Lemma 1.21. Let X and Y be Banach spaces and let A and V be linear operators from X to X and X to Y , respectively, such that D (A) ⊂ D (V ) and V is relatively bounded with respect to A. If, for each sequence (xn )∞ n=1 ⊂ ∞ D (A), the sequences (xn )∞ and (Ax ) converge weakly to zero, then also n n=1 n=1 (V xn )∞ converges weakly to zero. n=1

18


Proof. Let g be a bounded linear functional on Y . Define a linear functional f on the graph G (A) := (x, Ax) : x ∈ D (A) of A by f (x, Ax) = g(V x) for x ∈ D (A). Since V is A-bounded, we have f (x, Ax) 2 = | g(V x)|2 ≤ g 2 V x 2 ≤ g 2 max α , β 2 (x, Ax) 2 , that is, f is bounded on G (A). According to the Hahn-Banach-Theorem, there exists a bounded linear functional f on X ⊕ X , such that f = f on G (A). Consequently, (1.11) g(V x) = f (x, Ax) = f (x, Ax) = f (x, 0) + f (0, Ax) . ∞ ∞ If for some (xn )∞ n=1 ⊂ D (A) the sequences (x n ) n=1 and (Ax n ) n=1 converge ∞ weakly to zero, then, by equation (1.11), also (V xn )n=1 converges weakly ■ to zero.

The statements and proofs of the following theorem can be found in [Wei00, Satz 9.13] and [Hes69], respectively. Theorem 1.22. Let X , Y1 and Y2 be Banach spaces and let A and V be linear operators from X to Y1 and X to Y2 , respectively, such that D (A) ⊂ D (V ) and V is relatively compact with respect to A. Then V has A-bound 0 if (i) V is closable, or (ii) Y2 is reflexive and A is closable. Proof. By Remark 1.19, V is A-bounded. We prove the theorem by contradiction. Suppose that the A-bound of V is not 0. Then there exist an ε > 0 and a sequence (xn )∞ n=1 ⊂ D (A) such that for all natural numbers n (1.12)

V xn > n xn + ε Axn .

Set yn := xn / xn A . Then, by inequalities (1.7), (1.10) and (1.12), and since yn A = 1, n ∈ N, we have for n ≥ ε (1.13) max{α, β}= max{α, β} yn A ≥ V yn > n yn +ε A yn ≥ ε yn + A yn , where α and β are constants according to (1.7). Hence, by inequality (1.13), ∞ yn → 0 if n → ∞ and (yn )∞ n=1 and (A yn ) n=1 are bounded sequences. Thus, the A-compactness of V implies the existence of a subsequence (yn k )∞ of (yn )∞ n=1 k=1 ∞ such that (V yn k )k=1 converges to some z ∈ Y2 . In the following we show that, if either (i) or (ii) holds, then z = 0.


19

Let V be closable. Then, since (yn k )∞ converges to zero and (V yn k )∞ k=1 k=1 converges to z, it follows that z = 0. Now let Y2 be reflexive and A be closable. Since Y2 is reflexive and (A yn k )∞ is bounded, there exists, by [Wer07, Theorem III.3.7], a subsek=1 of (yn k )∞ such that (A yn k )∞ converges weakly to some quence (yn k )∞ k=1 l l =1 l l =1 v ∈ Y2 . We have (0, v) = w − lim (yn k , A yn k ) ∈ G (A) = G (A) l →∞

l

l

and hence v = 0. According to Lemma 1.21 it follows that also (V yn k )∞ l l =1 converges weakly to zero. Since we already know that (V yn k )∞ converges k=1 to z, it follows that z = 0. Altogether, since we have shown that (yn k )∞ and (V yn k )∞ converge k=1 k=1 ∞ to 0, (1.13) implies that so does (A yn k )k=1 . This leads to the contradiction 1 = yn k A = yn k + A yn k −→ 0,

k → 0.

■

We conclude the following: Remark 1.23. Let K ,[·, ·] be a Krein space and let A and V be linear operators in K ,[·, ·] such that V is A-compact. Then V has A-bound 0 if at least one of the operators A or V is closable.

Sufficient conditions for relatively bounded operators with relative bound 0 to be relatively compact can be found in [Hes69] and [Wei00, Section 9.2].

1.2.3 Stability of Self-Adjointness in Krein Spaces When a self-adjoint operator A in a Krein space is perturbed by a symmetric operator V which is relatively bounded with respect to A, then, in general, the perturbed operator A + V need not to be self-adjoint in the Krein space. In the Hilbert space case, a sufficient condition is that the A-bound of V is less than 1. This so called Kato-Rellich theorem extends to the case of Krein spaces. Theorem 1.24 (Kato-Rellich for Krein spaces). Let A be a self-adjoint operK ator in a Krein space ,[ · , · ] . If V is a symmetric and A-bounded operator in K , [·, ·] with A-bound less than 1, then A + V is self-adjoint in K , [·, ·] .

20


Proof. With the help of Lemma 1.8 and Remark 1.17 the proof follows from the ■ Hilbert space version of the theorem, see, e.g., [Kat95, Theorem V.4.3].

The Kato-Rellich theorem for Krein spaces gives a very important result regarding the structure of the spectrum of the perturbed operator A + V . If A + V is self-adjoint in K ,[·, ·] , its spectrum is symmetric with respect to the real axis. In particular, non-real eigenvalues occur as complex conjugate pairs.

1.3 Continuity of Separated Parts of the Spectrum In this section we consider relatively bounded perturbations of closed linear operators and their effect on the spectrum. If such an operator A 0 is perturbed, e.g., by a closed linear operator V which is A 0 -bounded with A 0 bound less than 1, then its spectrum cannot suddenly expand. In particular, we are interested in the case when the spectrum σ(A 0 ) consists only of countably many isolated eigenvalues or isolated parts. If, in this case, the linear operator A 0 is self-adjoint in a Hilbert and in a Krein space, then our main results state conditions which guarantee that also the spectrum of A 0 + V is real, even when A 0 + V is not self-adjoint in a Hilbert space.

1.3.1 Continuity of Resolvents In this paragraph we provide the results that will be the main tools in the following sections. While the “distance” between two bounded linear operators can be defined as the norm of their difference, the “distance” between two unbounded linear operators has to be measured in a different way. One possibility is to use the norm of the difference of their resolvents. This leads to the notion of convergence in the generalized sense considered below. To define convergence in the generalized sense, it is necessary to consider the gap between two closed linear operators, compare [Kat95, Paragraph IV.2].

1.3 Continuity of Separated Parts of the Spectrum

21

Definition 1.25. Let M and N be closed subspaces of a Banach space X . If M = {0}, set δ( M , N ) : =

sup

dist(x, N ),

x∈M , x =1

δ (M , N ) := max δ(M , N ), δ(N , M ) ;

if M = {0}, we define δ({0}, N ) = 0 for every N . Then δ (M , N ) is called the gap between M and N . Definition 1.26. Let X and Y be Banach spaces and let A 0 and A 1 be closed linear operators from X to Y . Since A 0 and A 1 are closed, their graphs G (A) and G (A 1 ) are closed linear subspaces of X × Y and we can define δ(A 0 , A 1 ) := δ G (A 0 ), G (A 1 ) , δ (A 0 , A 1 ) := δ G (A 0 ), G (A 1 ) = max δ(A 0 , A 1 ), δ(A 1 , A 0 ) .

Then δ (A 0 , A 1 ) is called the gap between A 0 and A 1 . If A ε , ε ∈ [0, 1], is a family of closed linear operators from X to Y , then A ε is said to converge to A 0 in the generalized sense if δ (A ε , A 0 ) → 0 for ε → 0. For the proof of the following theorem we refer the reader to [Kat95, Theorem IV.2.25]. Theorem 1.27. Let X be a Banach space and let A ε , ε ∈ [0, 1], be a family of closed linear operators in X such that ρ (A 0 ) = . In order that A ε converges to A 0 in the generalized sense, it is necessary that there exists some ε0 ∈ [0,1] such that each z ∈ ρ (A 0 ) also belongs to ρ (A ε ) for ε ∈ [0, ε0 ] and (1.14)

(A ε − z)−1 − (A 0 − z)−1 −→ 0,

ε → 0,

while it is sufficient that this is true for some z ∈ ρ (A 0 ). Corollary 1.28. Theorem 1.27 implies that if (1.14) holds for one z0 ∈ ρ (A 0 ), then (1.14) is true for all z ∈ ρ (A 0 ). Proposition 1.29. Let A and V be closed linear operators in a Banach space X such that D (A) ⊂ D (V ). Suppose that V is A-bounded such that (1.7) holds

22


with constants α ≥ 0 and β ∈ [0,1). Define the family of operators A ε := A + εV , 0 ≤ ε ≤ 1. If there exists a point z ∈ ρ (A 0 ) such that

α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1,

(1.15)

then z ∈ ρ (A ε ) and A ε converges to A 0 in the generalized sense. Proof. By (1.15), we have

V (A 0 − z)−1 ≤ α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1.

and thus, using Neumann’s series, we conclude I + εV (A 0 − z)−1 , 0 ≤ ε ≤ 1, has a bounded inverse with domain X . Thus also A ε − z = A 0 − z + εV = I + εV (A 0 − z)−1 (A 0 − z)

has a bounded inverse with domain X , and hence z ∈ ρ (A ε ) for 0 ≤ ε ≤ 1. By inequality (1.15) and since V is A-bounded with A-bound less than 1, we have

εV (A 0 − z)−1 x ≤ ε α (A 0 − z)−1 x + β A 0 (A 0 − z)−1 x for x ∈ K and 0 ≤ ε ≤ 1. Since (A 0 − z)−1 and A 0 (A 0 − z)−1 are bounded, the operator εV (A 0 − z)−1 is bounded with

εV (A 0 − z)−1 ≤ ε α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < ε −→ 0, ε → 0;

in particular, εV (A 0 − z)−1 < 1 for 0 ≤ ε ≤ 1. Using Neumann’s series, we conclude I + εV (A 0 − z)−1 has a bounded inverse and

I + εV (A 0 − z)−1 −1 ≤

1

.

1 − εV (A 0 − z)−1

Hence A ε − z = I + εV (A 0 − z)−1 (A 0 − z) has a bounded inverse. Using the second resolvent equation, we obtain

(A ε − z)−1 − (A 0 − z)−1 ≤ (A ε − z)−1 εV (A 0 − z)−1

(A 0 − z)−1

−1 ε V

−→ 0 ≤ (A 0 − z) 1 − ε V (A 0 − z)−1

if ε → 0.

■


23

1.3.2 Perturbation of Isolated Parts of the Spectrum In this paragraph we consider the situation where the spectrum σ(A) of a closed linear operator A in a Banach space X contains a bounded subset σ1 that is separated from the rest σ(A)\σ1 by a rectifiable closed curve Γ. For the case in which A is perturbed by an A-bounded operator V , we state conditions for the spectrum A + V to be likewise separated. We start with an introduction to the required notation. Definition 1.30. A closed Jordan curve is an oriented simple closed rectifiable curve γ : [a, b] → C, [a, b] ⊂ R. A subset Δ ∈ C is called Cauchy domain if there exist Δ i ⊂ C, i = 1, . . . , N, such that (i) Δ i = , Δ i open and connected for i = 1, . . . , N, (ii) Δ i ∩ Δ j = , i, j = 1, . . . , N, i = j, (iii) the boundary ∂Δ i of Δ i is a closed Jordan curve, i = 1, . . . , N.

Γ ⊂ C is called Cauchy contour if Γ is the oriented boundary of a Cauchy domain Δ, i.e., Γ = ∂Δ. Definition 1.31. Let A be a closed linear operator in a Banach space X . Then a bounded subset σ ⊂ σ(A) is called isolated part of σ(A) if σ and σ(A)\σ are closed. If Γ is a closed Cauchy contour such that σ ⊂ int Γ and (σ(A)\σ) ∩ (Γ ∪ int Γ) = , then σ(A) is said to be separated into two parts σ1 and σ2 by Γ. Definition 1.32. Let A be a closed linear operator in a Banach space X . Let M1 and M2 be two subspaces of X . Then A is said to be decomposed according to X = M1 ⊕ M2 (compare [Kat95, Paragraph III.5.6]) if there exists a projection P onto M1 along M2 such that P D (A) ⊂ D (A),

A M1 ⊂ M1 ,

A M2 ⊂ M2 .

In this case we define the closed linear operator A M1 in M1 by D (A M1 ) := D (A) ∩ M1 such that A M1 x := Ax ∈ M1 for x ∈ D (A M1 ). A M2 is defined accordingly. A proof of the following lemma can be found in [Kat95, Theorem III.6.17]. Lemma 1.33. Let A be a closed linear operator in a Banach space X and let its spectrum be separated into two parts σ1 and σ2 by a Cauchy contour Γ.

24


Then A can be decomposed according to X = M1 ⊕M2 such that σ(A M i ) = σ i , i = 1, 2, and A M1 is bounded in M1 . The projection P onto M1 = P X along M2 = (1 − P)X is given by 1 P := E(A, σ1 ) = − (A − z)−1 dz. (1.16) 2πi Γ Remark 1.34. The projection E(A, σ1 ) from (1.16) is called Riesz projection of A corresponding to σ1 (a detailed study of Riesz projections can be found in [GGK90]). Regarding the change of the spectrum under “small” perturbations we recall a theorem given in [Kat95], see Theorem IV.3.16 therein. Theorem 1.35. Let A 0 be a closed linear operator in a Banach space X and let the spectrum of A 0 be separated into two parts σ1 (A 0 ) and σ2 (A 0 ) by a Cauchy contour Γ. Let A 0 be decomposed according to X = M1 (A 0 ) ⊕M2 (A 0 ). Then there exists a δ > 0, depending on A 0 and Γ, such that any closed linear operator A 1 in X with δ (A 1 , A 0 ) < δ has spectrum σ(A 1 ) likewise separated by Γ into two parts σ1 (A 1 ) and σ2 (A 1 ), Γ ⊂ ρ (A 1 ). In the associated decomposition X = M1 (A 1 )⊕M2 (A 1 ), the subspaces M1 (A 1 ) and M2 (A 1 ) are isomorphic to M1 (A 0 ) and M2 (A 0 ), respectively. In particular, dim M i (A 1 ) = dim M i (A 0 ),

i = 1, 2,

and σ i (A 1 ) is non-empty if this is true for σ i (A 0 ), i = 1, 2. The decomposition X = M1 (A 1 ) ⊕ M2 (A 1 ) is continuous in A 1 in the sense that

E(A 1 , σ1 (A 1 )) − E(A 0 , σ1 (A 0 )) −→ 0 if δ (A 1 , A 0 ) → 0. Proof. According to [Kat95, Theorem IV.3.1], Γ ⊂ ρ (A 1 ) for every closed linear operator A 1 in X such that δ (A 1 , A 0 ) < δ with

2 −1/2

−1 1 δ = min 1 + | z|2 1 + (A 0 − z)−1 . 2 z∈Γ Hence σ(A 1 ) is separated by Γ into two parts σ1 (A 1 ) and σ2 (A 1 ). By Lemma 1.33, A 1 can be decomposed according to X = M1 (A 1 ) ⊕ M2 (A 1 ). We have M1 (A 1 ) = P X and M2 (A 1 ) = (1 − P)X , where P = E(A 1 , σ1 (A 1 )) is the Riesz projection of A 1 corresponding to σ1 (A 1 ). According to [Kat95, Theorem IV.3.15], for any z0 ∈ ρ (A 0 ) and ε > 0 there exists a δ0 > 0 such that


25

z ∈ ρ (A 1 ) and (A 1 − z)−1 − (A 0 − z0 )−1 < ε if | z − z0 | < δ0 and δ (A 1 , A 0 ) < δ0 . Further, since Γ is compact, (A 1 − z)−1 − (A 0 − z)−1 is uniformly small for z ∈ Γ if δ (A 1 , A 0 ) is sufficiently small. Hence

E(A 1 , σ1 (A 1 )) − E(A 0 , σ1 (A 0 )) ≤ 1

(A 1 − z)−1 − (A 0 − z)−1 dz 2π Γ

l(Γ)

(A 1 − z)−1 − (A 0 − z)−1 −→ 0 ≤ 2π

if δ (A 1 , A 0 ) → 0, where l(Γ) is the length of the Cauchy contour Γ. The isomorphism of M i (A 1 ) with M i (A 0 ), i = 1,2, and the fact that dim M i (A 1 ) = dim M i (A 0 ), i = 1, 2, now follow from [Kat95, Lemma I.4.10] which also holds in infinite dimensional Banach spaces. ■ The following two corollaries are special cases of Theorem 1.35 for relatively bounded perturbations. Corollary 1.36. Let A 0 be a closed linear operator in a Banach space X and let the spectrum of A 0 be separated into two parts σ1 (A 0 ) and σ2 (A 0 ) by a Cauchy contour Γ. Let A 0 be decomposed according to X = M1 (A 0 ) ⊕M2 (A 0 ). Further, let V be an A 0 -bounded operator in X and consider the family of operators A ε := A 0 + εV , 0 ≤ ε ≤ 1, in X . If there exists some ε0 ∈ [0, 1] such that each z ∈ ρ (A 0 ) belongs to ρ (A ε ) for ε ∈ [0, ε0 ] and

(A ε − z)−1 − (A 0 − z)−1 −→ 0, ε → 0, holds for some z ∈ ρ (A 0 ), then there exists some ε0 ∈ [0, ε0 ] such that σ(A ε ) is likewise separated by Γ into two parts σ1 (A ε ) and σ2 (A ε ), Γ ⊂ ρ (A ε ) and the results of Theorem 1.35 hold for 0 ≤ ε ≤ ε0 . Proof. By Theorem 1.27, for 0 ≤ ε ≤ ε0 , A ε converges to A 0 in the generalized sense, that is, δ (A ε , A 0 ) −→ 0 for ε → 0. Consequently, for any δ > 0 there exists an ε0 ∈ [0, ε0 ] such that δ (A ε , A 0 ) < δ if 0 ≤ ε ≤ ε0 and thus the assump■ tions of Theorem 1.35 are satisfied. The following corollary can be found in [Kat95, Theorem IV.3.18]. Corollary 1.37. Let A 0 , V be defined as in the preceding corollary and consider the operator A 1 := A 0 + V in X . If

(1.17) sup α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1, z ∈Γ

26


where α and β are constants according to (1.7), then the spectrum of A 1 is likewise separated into two parts σ1 (A 1 ) and σ2 (A 1 ), Γ ⊂ ρ (A 1 ) and the results of Theorem 1.35 hold. Proof. The proof is similar to the proof of Theorem 1.35. According to Proposition 1.29, inequality (1.17) guarantees that Γ ⊂ ρ (A 1 ). By Lemma 1.33, A 1 can be decomposed according to X = M1 (A 1 ) ⊕ M2 (A 1 ). Let A ε := A 0 + εV , 0 ≤ ε ≤ 1. As in the proof of Proposition 1.29, inequality (1.17) implies that (A ε − z)−1 , z ∈ Γ, depends continuously on ε for 0 ≤ ε ≤ 1. Thus the Riesz projection of A ε corresponding to σ1 (A ε ) is continuous in ε for 0 ≤ ε ≤ 1. The ■ last part of the proof is analogous to that of Theorem 1.35.

1.3.3 Perturbation of Spectra of Self-Adjoint Operators in Hilbert Spaces Compared to the preceding results, one can say much more about the stability of isolated parts of the spectrum if the unperturbed operator A 0 is self-adjoint in a Hilbert space. The reason for this is that the norm of the resolvent of a self-adjoint operator in a Hilbert space can be expressed in terms of the spectrum as follows. A proof of the following result may be found in [Kat95, Section V.3.8]. Proposition 1.38. Let A be a self-adjoint operator in a Hilbert space H . Then

(i) (A − z)−1 = sup |λ − z|−1 , z ∈ ρ (A), λ∈σ(A)

(ii) A(A − z)−1 = sup |λ||λ − z|−1 , z ∈ ρ (A). λ∈σ(A)

In the following, we consider families A ε = A 0 + εV , ε ∈ [0, 1], of closed linear operators where A 0 is self-adjoint in a Hilbert space H and V is an A 0 -bounded linear operator in H . We distinguish the following situations: (a) We consider one isolated eigenvalue (an infinite sequence of isolated eigenvalues, respectively) of the unperturbed operator A 0 . (b) We consider one isolated compact part (an infinite sequence of isolated compact parts, respectively) of the spectrum of the unperturbed operator A 0 .


27

Here and in the sequel for δ > 0 and z0 ∈ C or M0 ⊂ C we denote by Bδ (z0 ) := z ∈ C : | z − z0 | < δ , Bδ (z0 ) := z ∈ C : ∀ z0 ∈ M0 | z − z0 | < δ

the δ neighbourhood of z0 and M0 , respectively; we also define Z∗ := { x ∈ Z : x = 0}. In situation (a) we obtain the following result. Theorem 1.39. Let A 0 be a self-adjoint operator and V an A 0 -bounded operator in a Hilbert space H with A 0 -bound less than 1/2. Define the family of operators A ε := A 0 + εV , ε ∈ [0,1]. (i) Let λ0 ∈ R be an isolated eigenvalue of A 0 with multiplicity m < ∞ and set δ := dist λ0 , σ(A 0 )\{λ0 } . If (1.7) holds with constants α ≥ 0 and β ∈ [0, 1/2) such that 1 1 α + β δ + |λ0 | < , δ 2

(1.18)

then σ(A 1 ) ∩ Bδ/2 (λ0 ) consists of a finite system of isolated eigenvalues with total multiplicity m. (ii) Let A 0 have discrete spectrum consisting of eigenvalues · · · < λ0−2 < λ0− 1 < λ01 < λ02 < · · · with multiplicities m n < ∞, n ∈ Z∗ , and set δn := dist λ0n , σ(A 0 )\{λ0n } , n ∈ Z∗ . If (1.7) holds with constants αn ≥ 0 and βn ∈ [0, 1/2), n ∈ Z∗ , such that

(1.19)

γ := sup

n∈Z∗

1 αn + βn δn + |λ0n | < ∞, δn

then the spectrum of A ε is discrete for all ε ∈ [0, ε0 ], where ε0 ∈ (0,1] has to be chosen such that ε0 < 1/(2γ). More precisely, σ(A ε ) ∩ Bδn /2 (λ0n ) consists of a finite system of isolated eigenvalues with total multiplicity m n for all ε ∈ [0, ε0 ] and n ∈ Z∗ . 0 the circle Proof. (i). Let Γ be the positively oriented curve along with center λ 0 0 and radius δ/2. Then Γ ⊂ ρ (A 0 ), {λ } ⊂ int Γ and Γ ∪ int Γ ∩ σ(A 0 )\{λ } = .

28


If z ∈ Γ, then, by the choice of Γ, |λ − z|−1 ≤ 2/δ, λ ∈ σ(A 0 ). Furthermore, since |λ0 − z| = δ/2, we obtain that for λ ∈ σ(A 0 ) |λ||λ − z|−1 ≤ |λ − z| + | z| |λ − z|−1 ≤ 1 + | z − λ0 | + |λ0 | |λ − z|−1 ≤ 2+

2|λ0 | . δ

By Proposition 1.38, since A 0 is self-adjoint in the Hilbert space H , inequality (1.17) is satisfied if 2 2|λ0 | α +β 2+ < 1, δ δ or, equivalently, (1.18) holds. The application of Corollary 1.37 completes the proof. (ii). By (1.19), the assumptions of part (i) are satisfied for A 0 and εV with ε ∈ [0, ε0 ] for every eigenvalue λ0n , n ∈ Z∗ , if we choose ε0 such that ε0 sup

n∈Z∗

1 1 αn + βn δ n + |λ0n | < . δn 2

■

The following theorem deals with situation (b), where isolated compact parts of the spectrum of the unperturbed operator are considered. Theorem 1.40. Let A 0 be a self-adjoint operator and V an A 0 -bounded operator in a Hilbert space H with A 0 -bound less than 1/2. Define the family of operators A ε := A 0 + εV , ε ∈ [0, 1]. (i) Let σ0 be an isolated part of σ(A 0 ) such that σ0 = σ(A 0 ) ∩ [λ− , λ+ ] with − + λ , λ ∈ R and set δ := dist σ0 , σ(A 0 )\σ0 . If (1.7) holds with constants α ≥ 0 and β ∈ [0, 1/2) such that (1.20)

1 1 α + β δ + max |λ− |, |λ+ | < , δ 2

then ∂Bδ/2 ([λ− , λ+ ]) ⊂ ρ (A ε ) and σε := σ(A ε ) ∩ Bδ/2 ([λ− , λ+ ]) is an isolated part of σ(A ε ) for all ε ∈ [0,1]. Furthermore, dim E(A 0 , σ0 )K = dim E(A ε , σε )K for ε ∈ [0,1].


(ii) Let σ(A 0 ) =

n∈Z∗

29

σn,0

+ ∗ − + with σn,0 = σ(A 0 ) ∩ [λ− n , λ n ], n ∈ Z , where λn , λ n ∈ R, and + − + − + − + · · · < λ− −2 ≤ λ−2 < λ−1 ≤ λ−1 < λ1 ≤ λ1 < λ2 ≤ λ2 < · · · .

Define δn := dist σn,0 , σ(A 0 )\σn,0 , n ∈ Z∗ . If (1.7) holds with constants αn ≥ 0 and βn ∈ [0,1/2), n ∈ Z∗ , such that

(1.21)

γ := sup

n∈Z∗

1 + αn + βn δ n + max |λ− < ∞, n |, |λ n | δn

+ − + then ∂Bδn /2 ([λ− n , λn ]) ⊂ ρ (A ε ) and σ n,ε := σ(A ε ) ∩ Bδ n /2 ([λ n , λn ]) is an iso∗ lated part of σ(A ε ) for all n ∈ Z , ε ∈ [0, ε0 ], and

σ(A ε ) =

n∈Z∗

σn,ε ,

ε ∈ [0, ε0 ],

where ε0 ∈ (0, 1] has to be chosen such that ε0 < 1/(2γ). Furthermore, dim E(A 0 , σn,0 )K = dim E(A ε , σn,ε )K for ε ∈ [0, ε0 ] and n ∈ Z∗ . Proof. (i). Let Γ ⊂ ρ (A 0 ) be the positively oriented curve along ∂Bδ/2 ([λ− , λ+ ]). + −1 ≤ |λ z − z|−1 = δ/2, For all z ∈ Γ there exists a λ z ∈ [λ− 0 , λ0 ] such that |λ − z| λ ∈ σ(A 0 ). For any z ∈ Γ and any λ ∈ σ(A 0 ) we have, as above, |λ||λ − z|−1 ≤ 2 +

+ 2 max |λ− 2|λ z | n |, |λ n | ≤ 2+ . δ δ

Since inequality (1.20) satisfies inequality (1.17), the statement follows from Corollary 1.37. (ii). By (1.21), the assumptions of part (i) are satisfied for A 0 and εV with ε ∈ [0, ε0 ] for every isolated compact part σn,0 , n ∈ Z∗ , if we choose ε0 such that 1 1 + ε0 sup αn + βn δn + max |λ− | , | λ | < . ■ n n 2 n∈Z∗ δ n Remark 1.41. If in Theorem 1.39 (ii) and Theorem 1.40 (ii) A 0 is semi+ − + bounded, λ01 < λ02 < · · · or λ− 1 ≤ λ1 < λ2 ≤ λ2 < · · · , it follows that σ(A 0 + εV ) is bounded from below for ε ∈ [0, ε0 ].

30


Remark 1.42. Applying similar arguments as in Theorem 1.40 for the case of one spectral gap (a, b) in the spectrum of a self-adjoint operator A 0 , one may obtain the following result, compare [Cue]. Let V be A 0 -bounded with A 0 -bound less than 1. If (1.22)

1 1 α + β max |a|, | b| < , δ 2

where α and β are constants according to (1.7) and δ := b − a, then the open strip I + iR, where the interval I is given by I := a + α + β max |a|, | b| , b − α + β max |a|, | b| , is contained in the resolvent set ρ (A 1 ) of A 1 := A 0 + V . Note that the interval I is non-empty if inequality (1.22) is satisfied. As in the proof of Theorem 1.40 one can show that

α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1

if z ∈ I, and hence z ∈ ρ (A 1 ) by [Kat95, Theorem IV.3.17].

1.3.4 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces In this section we consider the case where A 0 and V are self-adjoint and symmetric in a Krein space, respectively. We combine Theorems 1.39 and 1.40 on the stability of isolated parts of the spectrum with a stability result for uniformly positive subspaces of a Krein space. This allows us to establish conditions guaranteeing that the spectrum of A 0 + V is real provided the spectrum of A 0 is real. The following stability result for uniformly definite subspaces of a Krein space was proved in [LT04, Theorem 3.1]. This theorem has already been applied for bounded perturbations V in [LT04, Theorem 3.1] and it is also fundamental for unbounded perturbations considered below. Theorem 1.43. Let A ε , 0 ≤ ε ≤ 1, be a family of self-adjoint operators in a Krein space K ,[·, ·] such that for one (and hence for all) z ∈ ρ (A ε ) the resolvents (A ε − z)−1 depend continuously on ε in the operator norm for 0 ≤ ε ≤ 1. Let σ0 be an isolated part of the spectrum σ(A 0 ) of A 0 such that σ0 = σ∗0 , and let Γ be a Cauchy contour surrounding σ0 such that Γ = Γ∗ . Suppose


31

that σ(A ε ) ∩ Γ = for all 0 ≤ ε ≤ 1 and set σε := σ(A ε ) ∩ int Γ. If the subspace E(A 0 , σ0 )K is uniformly positive (uniformly negative, respectively), then for all 0 ≤ ε ≤ 1 the subspaces E(A ε , σε )K are uniformly positive (uniformly negative, respectively) and the sets σε are real. As in the preceding paragraph the following two theorems cover situations (a) (isolated eigenvalues) and (b) (isolated compact parts of the spectrum), respectively. Theorem 1.44. Let A 0 be a self-adjoint operator in a Krein space K , [·, ·] K , (·, ·) . Further, let such that A 0 is also self-adjoint in the Hilbert space V be a symmetric operator in the Krein space K , [·, ·] which is A 0 -bounded with A 0 -bound less than 1/2 and define the family of operators A ε := A 0 + εV , ε ∈ [0, 1].

(i) Let λ0 ∈ R be an isolated eigenvalueof A 0 which isof definite type with multiplicity m < ∞ and set δ := dist λ0 , σ(A 0 )\{λ0 } . If (1.7) holds with constants α ≥ 0 and β ∈ [0,1/2) such that 1 1 α + β δ + |λ0 | < , δ 2

(1.23)

then σ(A 1 )∩Bδ/2 (λ0 ) consists of a finite system of isolated and real eigenvalues with total multiplicity m which are of the same type as λ0 . (ii) Let A 0 have discrete spectrum consisting of eigenvalues · · · < λ0−2 < λ0−1 < λ01 < λ02 < · · · of definite type with multiplicities m n < ∞, n ∈ Z∗ , and define δn := dist λ0n , σ(A 0 )\{λ0n } , n ∈ Z∗ . If (1.7) holds with constants αn ≥ 0 and βn ∈ [0,1/2), n ∈ Z∗ , such that

(1.24)

γ := sup

n∈Z∗

1 αn + βn δn + |λ0n | < ∞, δn

then the spectrum of A ε is discrete and consists of real eigenvalues of definite type for all ε ∈ [0, ε0 ], where ε0 ∈ (0, 1] has to be chosen such that ε0 < 1/(2γ). More precisely, σ(A ε ) ∩ Bδn /2 (λ0n ) consists of a finite system of real isolated eigenvalues with total multiplicity m n which are of the same type as λ0n for all ε ∈ [0, ε0 ] and n ∈ Z∗ . Proof. (i). For the proof we apply Corollary 1.37 and Theorem 1.43. The fam ily of operators A ε , 0 ≤ ε ≤ 1, is self-adjoint in the Krein space K ,[·, ·] by Theorem 1.24. Let Γ be the positively oriented curve along the circle with

32


center λ0 and radius δ/2. Then Γ ⊂ ρ (A 0 ) separates the spectrum of A 0 into the two parts λ0 and σ(A 0 )\ λ0 . As we have seen in the proof of Theorem 1.39, by inequality (1.23), the assumptions of Corollary 1.37 are satisfied. Thus the spectrum of A ε = A 0 + εV is also separated into two parts by Γ, where Γ itself is running in ρ (A ε ). According to Corollary 1.37, the eigenvalue λ0 of A 0 splits at most into a finite system of isolated eigenvalues of A ε enclosed by Γ which have total multiplicity m. By Proposition 1.29, A ε converges to A 0 in the generalized sense. Now Theorem 1.43 can be applied. If we define σε := σ(A ε ) ∩ int Γ, 0 ≤ ε ≤ 1, then σε = σ∗ε for all 0 ≤ ε ≤ 1. This in is due to the fact that spectrum of the operator A ε , which is self-adjoint ∗ symmetric with respect to the real axis and that Γ = Γ . We note K , [·, ·] , is that σ0 = λ0 . Since, by assumption, λ0 is of definite type, E(A 0 , σ0 )K is either uniformly positive or uniformly negative. According to Theorem 1.43 the subspace E(A ε , σε )K is of the same type as E(A 0 , σ0 )K and the set σε is real for all 0 ≤ ε ≤ 1. Hence σε consists of a finite system of real eigenvalues of A ε with total multiplicity m and which is of the same type as λ0 . (ii). By (1.24), the assumptions of part (i) are satisfied for A 0 and εV with ε ∈ [0, ε0 ] for every eigenvalue λ0n , n ∈ Z∗ , if we choose ε0 such that ε0 sup

n∈Z∗

1 1 αn + βn δ n + |λ0n | < . δn 2

■

Remark 1.45. For a special class of differential operators the last corollary has also been proved in a paper by E. Caliceti and S. Graffi (see [CG05]; see also [Cal05] and [CCG06]), compare Theorem 3.20 and Remark 3.21 below. They consider operators induced by the differential expression Aε = −

d2 + P + ε iQ, dx2

ε ∈ [0,1],

in L2 (R), where P and Q are multiplication operators by real polynomials P and Q. In Section 3.3 we show that the conditions of Theorem 1.44 (ii) are satisfied in this special case with A0 = −

d2 +P dx2

and

V = iQ.

K , [·, ·] Theorem 1.46. Let A 0 be a self-adjoint operator in a Krein space such that A 0 is also self-adjoint in the Hilbert space K ,(·, ·) . Further, let V be a symmetric operator in the Krein space K , [·, ·] which is A 0 -bounded


33

with A 0 -bound less than 1/2 and define the family of operators A ε := A 0 + εV , ε ∈ [0, 1]. (i) Let σ0 be an isolated part of σ(A 0 ) such that E(A 0 , σ0 )K is uniformly definite with σ0 = σ(A 0 ) ∩ [λ− , λ+ ], where λ− , λ+ ∈ R. Define the distance δ := dist σ0 , σ(A 0 )\σ0 . If (1.7) holds with constants α ≥ 0 and β ∈ [0, 1/2) such that 1 1 α + β δ + max |λ− |, |λ+ | < , δ 2

then the set σε := σ(A ε ) ∩ Bδ/2 ([λ− , λ+ ]) is real and an isolated part of σ(A ε ) for all ε ∈ [0, 1]. Furthermore, dim E(A 0 , σ0 )K = dim E(A ε , σε )K for ε ∈ [0, 1]. (ii) Let σ(A 0 ) =

n∈Z∗

σn,0

+ ∗ − + ∗ with σn,0 = σ(A 0 ) ∩ [λ− n , λ n ], n ∈ Z , where λn , λ n ∈ R, n ∈ Z , and + − + − + − + · · · < λ− −2 ≤ λ−2 < λ−1 ≤ λ−1 < λ1 ≤ λ1 < λ2 ≤ λ2 < · · · .

Suppose that E(A 0 ,σn,0 )K is uniformly definite for n ∈ Z∗ . Define the distances δn := dist σn,0 , σ(A 0 )\σn,0 , n ∈ Z∗ . If (1.7) holds with constants αn ≥ 0 and βn ∈ [0,1/2), n ∈ Z∗ , such that γ := sup

n∈Z∗

1 + αn + βn δ n + max |λ− | , | λ | < ∞, n n δn

+ then the sets σn,ε := σ(A ε ) ∩ Bδn /2 ([λ− n , λn ]) are real and isolated parts of σ(A ε ) and σ(A ε ) = σn,ε ⊂ R, ε ∈ [0, ε0 ],

n∈Z∗

where ε0 ∈ (0, 1] has to be chosen such that ε0 < 1/(2γ). Furthermore, dim E(A 0 , σn,0 )K = dim E(A ε , σn,ε )K for ε ∈ [0, ε0 ] and n ∈ Z∗ . Proof. With the help Theorem 1.40, the proof is analogous to the proof of Theorem 1.44. ■ When, in contrast to the two preceding theorems, the unperturbed operator A 0 is only self-adjoint in a Krein space K ,[·, ·] , then we have the following result.

34


Theorem 1.47. Let A 0 be a self-adjoint operator in a Krein space K , [·, ·] with discrete real spectrum consisting of eigenvalues · · · < λ0−2 < λ0−1 < λ01 < λ02 < · · · of definite type. Let Γn ⊂ ρ (A 0 ), n ∈ Z∗ , be a closed Jordan curve λ0n of A 0 such that Γn = Γ∗n and Γ n ∪ int (Γn ) ∩ surrounding the eigenvalue σ(A 0 )\{λ0n } = . Set Γ := n∈Z∗ Γn . Let V be a symmetric operator in the Krein space K ,[·, ·] which is A 0 -bounded. If (1.7) holds with constants α ≥ 0 and β ∈ [0, 1) such that

(1.25) γ := sup α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < ∞, z ∈Γ

then the spectrum of A 0 + εV is discrete and real and consists of eigenvalues of definite type for all ε ∈ [0, ε0 ], where ε0 ∈ (0,1] has to be chosen such that ε0 < 1/(2γ). More precisely, if m n denotes the multiplicity of the eigenvalues λ0n of A 0 , n ∈ Z∗ , then under the perturbation εV the eigenvalues λ0n of A 0 split into a finite system of isolated eigenvalues with total multiplicity m n which are of the same type as λ0n . Proof. Since A 0 is not assumed to be self-adjoint in a Hilbert space, we cannot use Theorem 1.39, but verify the assumptions of Corollary 1.37 directly. In fact, for each n ∈ Z∗ the spectrum of A 0 is separated into two parts by Γn . If we choose ε0 such that

1 ε0 sup α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < , 2 z∈Γ

then the assumptions of Corollary 1.37 are satisfied for A 0 and εV with ε ∈ [0, ε0 ] for every λ0n , n ∈ Z∗ . Now the results follow as in the proof of Theorem 1.44. ■ Remark 1.48. If V is bounded, then α = V , β = 0 and [LT04, Corollary 3.4] is a special case of Theorem 1.47 (see also [LT06]).

Chapter 2 Relatively Form-Bounded Perturbations in Krein Spaces In Chapter 1 we considered perturbations A 0 + V of a closed linear operator A 0 by a linear operator V for which A 0 + V is defined as an operator sum. In this chapter we consider operators A 0 and V for which the sum is defined only by means of quadratic forms. We will extend the main results from Chapter 1 to this more general case.

2.1 Stability Theorems As in the case of relatively bounded perturbations, a number of important properties of linear operators are preserved under relatively form-bounded perturbations. We recall the basic definitions and give a short review on quadratic forms and associated operators. A detailed study of quadratic forms as well as of relatively form-bounded and relatively form-compact operators can be found in [Kat95], [RS80], [RS75], [RS78] and [EE87].

2.1.1 Accretive and Sectorial Operators Definition 2.1. (i) Let K ,[·, ·] be a Krein space and let A be a linear operator in K ,[·, ·] . We define the set W [∗] (A) ⊂ C by W [∗] (A) := [Ax, x] : x ∈ D (A), x = 1 . (ii) A symmetric operator A in K called bounded from below , [·, ·] [∗is (in the Krein space K , [·, ·] ) if W ] (A) (which is a subset of R) is bounded from below, that is, if there exists some γ ∈ R such that

[Ax, x] ≥ γ x 2 ,

x ∈ D (A).

If γ = 0, then A is said to be non-negative (in the Krein space K ,[·, ·] ). J. Nesemann, PT-Symmetric Schrödinger Operators with Unbounded Potentials, DOI 10.1007/978-3-8348-8327-8_3, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

36

Chapter 2 Relatively Form-Bounded Perturbations in Krein Spaces

(iii) The linear operator A is said to be accretive (in the Krein space K , [·, ·] ) if W [∗] (A) is a subset of the right half-plane, i.e., if Re [Ax, x] ≥ 0,

x ∈ D (A).

Further, A is called quasi-accretive (in the Krein space K ,[·, ·] ) if A + γ is accretive for some γ > 0.

(iv) In the case when A isadditionally closed, A is said to be m-accretive (in the Krein space K ,[·, ·] ) if it admits no non-trivial accretive extensions. If A + γ is m-accretive for some γ > 0, then A is called quasim-accretive (in the Krein space K , [·, ·] ). (v) The linear operator Ais called sectorially-valued or simply sectorial (in the Krein space K , [·, ·] ) if W [∗] (A) is a subset of a sector π z ∈ C : Re z ≥ γ, | arg(z − γ)| ≤ θ < , 2 for some γ ∈ R and 0 ≤ θ < π/2. We shall call γ a vertex and θ a semiis angle of the sectorial operator A. Note that a sectorial operator quasi-accretive. A is called m-sectorial (in the Krein space K , [·, ·] ) if it is sectorial and quasi-m-accretive. Remark 2.2. The above notations are sometimes also referred to as J-nonspace negative, J-accretive etc. because, e.g., A is non-negative in the Krein K , [·, ·] if and only if J A is non-negative in the Hilbert space K , (·, ·) . If the (classical) numerical range of A in a Hilbert space is denoted by W(A), then W [∗] (A) = W(J A) and vice versa since [x, y] = (J x, y), x, y ∈ K . Remark 2.3. The set W [∗] (A) differs from the set called Krein space numer ical range which was defined in [LTU96] as [Ax, x] : x ∈ D (A), [x, x] = 1 .

2.1.2 Quadratic Forms and Associated Operators Definition 2.4. Let X be a vector space over K (K = R or C). A sesquilinear form is a map a : D × D → C such that (i) a α x1 + β x2 , y = αa x1 , y + βa x2 , y, (ii) a x, α y1 + β y2 = αa x, y1 + βa x, y2 , where D is a subspace of X and x1 , x2 , y1 , y2 ∈ D , α, β ∈ K. Further, a : D → C, a x = a x, x is called quadratic form associated with a x, y. The subspace D = D (a) is called domain of the sesquilinear or quadratic form a. When there


37

is no possibility of confusion we call the sesquilinear form a x, y or quadratic form a x simply a form. Definition 2.5. (i) Let X be a vector space over K (K = R or C). The sesquilinear form a∗ defined by

a∗ x, y := a y, x,

x, y ∈ D (a∗ ) = D (a),

is called the adjoint form of a. A form a : D (a) × D (a) → C is said to be symmetric if a∗ = a, that is, if

a x, y = a y, x,

x, y ∈ D (a);

in this case the quadratic form a is real valued. (ii) A symmetric form a is called bounded from below if there exists a γ ∈ R such that a ≥ γ, that is,

a x ≥ γ x 2 ,

x ∈ D (a).

If γ ≥ 0, then a is called non-negative. The notions of boundedness from above and non-positiveness may be defined analogously. (iii) For a quadratic form a we define the numerical range W(a) by W(a) := a x : x ∈ D (a), x = 1 . (iv) The quadratic form a is called sectorially bounded from the left or simply sectorial if W(a) is a subset of a sector π z ∈ C : Re z ≥ γ, | arg(z − γ)| ≤ θ < , 2 for some γ ∈ R and 0 ≤ θ < π/2. We call γ a vertex and θ a semi-angle of the form a. In the following we define closed and closable forms. First we have to introduce the notion of a-convergence. Definition 2.6. (i) Let a be a form in a Banach space X . A sequence ⊂ D ( a ) is called a-convergent (to x ∈ X ) if xn → x and (xn )∞ n=0 a xn − xm → 0 for m, n → ∞. (ii) If a is sectorial, then a is said to be closed if for any a-convergent sequence (xn )∞ n=0 ⊂ D (a) we have x ∈ D (a) and a x n − x → 0 for n → ∞.

38


(iii) A sectorial form a is said to be closable if it has a closed extension; in this case the smallest closed extension a is called the closure of a. Example 2.7. Let A be a linear operator in a Krein space K ,[·, ·] . Then A induces a sesquilinear form a on K by means of

a x, y := [Ax, y],

x, y ∈ D (a) = D (A).

Clearly, if A is symmetric in K , [·, ·] , then the form a is symmetric. If A is sectorial with vertex γ, then a is sectorial with vertex γ.

Definition 2.8. Let a be a closed sectorial form in a Krein space K with vertex γ and define x a := (Re a − γ + 1)1/2 x,

x ∈ D (a).

A subspace D of K is said to be a core of a if D is dense in the Banach space D (a), · a (compare, e.g., [Kat95, Section VI.1.3]). The following theorem is a generalization of the well-known first representation theorem for quadratic forms in a Hilbert space (see [Kat95, Paragraph VI.2]) to quadratic forms in a Krein space (see [Fle99, Theorem 1]). Theorem 2.9 (The first representation theorem). Let K ,[·, ·] be a Krein space and let a be a sectorial sesquilinear form in K which is closed with respect to the Hilbert space topology of K . Then there exists an m-sectorial operator A in K ,[·, ·] such that

(i) D (A) ⊂ D (a) and

a x, y = [Ax, y],

x ∈ D (A), y ∈ D (a);

(ii) D (A) is a core of a; (iii) if x ∈ D (a), z ∈ K and

a x, y = [z, y] holds for every y belonging to a core of a, then x ∈ D (A) and Ax = z. Proof. According to the Hilbert space version of the first representation theorem, see, e.g., [Kat95, Theorem VI.2.1], there exists an m-sectorial operator


39

a in the Hilbert space (K , (·, ·) such that above conditions (i) to (iii) hold A a with [·, ·] replaced by (·, ·) in (i) and (iii). If we define for A

(2.1)

a , A a := J A

then A a is m-sectorial in the Krein space K ,[·, ·] . Now (i) follows immedia x, y) = [J A a x, y] = [A a x, y] for x ∈ D (A a ) = D ( A a ) and y ∈ D (a). ately since ( A a ), and so is statement (iii); for Statement (ii) is obvious because D (A a ) = D ( A z ∈ K we use the identity (z , y) = [J z , y] = [z, y], with z := J z , and hence a x = J z = z for x ∈ D (A a ). Aa x = J A ■

The proofs of the following results are similar to the Hilbert space case, compare [Kat95, Paragraph VI.2]. Corollary 2.10. If a form a0 is induced by the operator A a in the Krein space in Theorem 2.9 by a0 x, y = [A a x, y], x, y ∈ D (a0 ) = D (A a ), then the form a in Theorem 2.9 is the closure of a0 . Proof. Corollary 2.10 is immediate from statement (ii) of Theorem 2.9.

■

Corollary 2.11. If B is a linear operator such that D (B) ⊂ D (a) and a x, y = [Bx, y] for every x ∈ D (B) and every y belonging to a core of a, then B ⊂ A a . Proof. Corollary 2.11 is a direct consequence of statement (iii) of Theorem 2.9. If x ∈ D (B) ⊂ D (a), then we have a x, y = [z, y] with z := Bx. By (iii) of Theo■ rem 2.9, x ∈ D (A a ) and A a x = z = Bx, i.e., B ⊂ A a . Proposition 2.12. The mapping a → A = A a is a one-to-one correspondence between the set of all densely defined, closed forms and the set of all sectorial m-sectorial operators in the Krein space K , [·, ·] . The form a is bounded if and a is symmetric if and only if A is self-adjoint and only if A is bounded, in the Krein space K , [·, ·] . Proof. First we show that the mapping a → A = A a is injective. Let a1 and a2 be densely defined, closed and sectorial sesquilinear forms. Then, by (i) of Theorem 2.9, A a1 = A a2 implies that a1 = a2 restricted to D (A a1 ) = D (A a2 ). If we define the forms a10 x, y = [A a1 x, y] with D (a10 ) = D (A a1 ) and a20 x, y = [A a2 x, y] with D (a20 ) = D (A a2 ), then

a1 = a10 = a20 = a2 .

40


It remains to show that the mapping a → A = A a is surjective. Let A be an m-sectorial operator in K . As we have seen in Corollary 2.10, we obtain a densely defined, closed sectorial form a as the closure of the densely defined sectorial form

a0 x, y = [Ax, y],

D (a0 ) = D (A).

Denote the m-sectorial operator of Theorem 2.9 by A a . Then, by Corol lary 2.11, A a ⊃ A. Hence, since A a and A are m-sectorial in K ,[·, ·] , we ■ obtain A a = A. Remark 2.13. The uniquely defined operator A a in Theorem 2.9 is called the m-sectorial or simply the operator associated with a in the operator Krein space K ,[·, ·] . Vice versa, the form a is called the form associated with A. Next we present the Krein space version of the second representation theorem which can be found in [Kat95], see Theorem VI.2.23 therein. Theorem 2.14 (The second representation theorem). Let a be a densely de fined, closed and non-negative sesquilinear form in a Krein space K , [·, ·] . Let A = A a be the associated self-adjoint operator in K ,[·, ·] . Then we have D (a) = D (J A)1/2 and

a x, y = (J A)1/2 x, (J A)1/2 y ,

x, y ∈ D (a).

Furthermore, a subspace D of D (a) is a core of a if and only if it is a core of (J A)1/2 . Proof. The proof follows immediately from the Hilbert space version of the theorem, compare [Kat95, Theorem VI.2.23] if we note that the operator as■ sociated with a in the Hilbert space K ,(·, ·) is J A, compare (2.1). 1/2 Remark in the 2.15. The unique m-accretive square root (J A) Hilbert space K , (·, ·) of J A is self-adjoint and non-negative in K ,(·, ·) such that 2 (J A)1/2 = (J A) and D (J A) (= D (A)) is a core of (J A)1/2 , see [Kat95, Paragraph V.3.11].


41

Definition 2.16. Let A be a densely defined, sectorial operator in a Krein space K , [·, ·] and let a0 be the closable form associated with A by

a0 x, y := [Ax, y],

x, y ∈ D (a0 ) = D (A).

Then the operator AF := A a0 ⊃ A is called Friedrichs extension of A in the Krein space K , [·, ·] . Remark 2.17. If A is a symmetric in a Krein space K , [·, ·] such operator that A is bounded from below in K , [·, ·] , then A is sectorial and hence the Friedrichs extension A F of Aexists. By [Kat95, Theorem VI.2.6], J A F is selfadjoint in the Hilbert space K , (·, ·) , that is, A F is self-adjoint in the Krein space K , [·, ·] .

Remark 2.18. Since an m-sectorial operator in a Krein space has no proper sectorial extension (see Definition 2.1), the Friedrichs extension of an msectorial operator A in a Krein space is A itself. Definition 2.19. Let a0 and v be closed and sectorial forms in a Krein space K , [·, ·] . By [Kat95, Theorem VI.1.31], the same is true for their sum a1 = a0 + v. Thus if a1 is densely defined, then the m-sectorial operators A 1 , A 0 and V associated with a1 , a0 and v, respectively, are defined. A 1 is the “sum” of A 0 and V in a generalized sense, which is called the form sum of A 0 and V and indicated by writing A 1 = A 0 V . Remark 2.20. (i) If any two operators A 0 and V are self-adjoint and bounded from below in a Krein space K , [·, ·] , then the associated forms a0 and v exist. In this case the form sum A 1 of A 0 and V is the operator associated with the form a1 = a0 + v provided a1 is densely defined. The requirement of a1 being densely defined is weaker than the condition for the ordinary operator sum S := A 0 + V to be densely defined. In some cases it may not be possible to define the operator sum, e.g., D (A 1 ) ∩ D (A 0 ) = {0} may occur, while the form sum is defined (an example can be found in [Wei00, Übung 4.17]). Even when S is densely defined it may not be self-adjoint in K , [·, ·] or have a selfadjoint closure. From (iii) of Theorem 2.9 it follows that A 1 = A 0 V is an extension of the operator sum A 0 + V .

42


(ii) If A 0 and V are self-adjoint and bounded from below in K , [·, ·] such that A 0 + V is densely defined, the Friedrichs extension A F of A 0 + V exists. In general, however, it is not true that A F = A 0 V , see [Kat95, Example VI.2.19].

For the preceding remark compare also [Kat95, Section VI.2.5] and [EE87, Section IV.2].

2.1.3 Relatively Form-Bounded and Relatively Form-Compact Operators The first representation theorem, Theorem 2.9, gives a one-to-one correspondence between densely defined, closed sectorial forms and m-sectorial operators in a Krein space K ,[·, ·] . Using this correspondence, we extend the notion of relative boundedness for forms (see [Kat95, Section VI.1.6 ff.] and Definition 2.21 below) to the associated m-sectorial operators in a Krein space K , [·, ·] . Definitions of relatively form-bounded operators in Hilbert spaces may be found in [Tes09, Section 6.5], [RS75, Section X.2] and [BH04]. The following definition was given in [Kat95, Section VI.1.6]. Definition 2.21. Let a and v be sesquilinear forms in a Krein space K , [·, ·] . Suppose that a is sectorial. If D (a) ⊂ D (v) and there exist non-negative constants α and β such that

(2.2)

|v x| ≤ α x 2 + β|a x|,

x ∈ D (a),

then v is called relatively bounded with respect to a. The greatest lower bound β0 of all possible constants β in (2.2) is called a-bound of v. Remark 2.22. If in Definition 2.21 both a and v are sectorial and closable, then, according to [Kat95, Paragraph VI.1.6], inequality (2.2) also holds for their closures with the same constants α and β. A be a self-adjoint and V a symmetric operator in a Definition 2.23. Let Krein space K ,[·, ·] such that A and V are bounded from below in K , [·, ·] . Let a and v be the forms associated with A and with the Friedrichs extension VF of V , respectively. If D (a) ⊂ D (v) and there exist non-negative constants α and β such that (2.3)

|v x| ≤ α x 2 + β|a x|,

x ∈ D (a),


43

then V is called relatively form-bounded with respect to A in the Krein space K , [·, ·] . The greatest lower bound β0 of all possible constants β in (2.3) is called relative form-bound of V with respect to A. Remark 2.24. If A and V are not only bounded from below but non-negative in the Krein space K , [·, ·] , then inequality (2.3) is equivalent to

(JVF )1/2 x 2 ≤ α x 2 + β (J A)1/2 x 2 , x ∈ D (J A)1/2 , with D (J A)1/2 ⊂ D (JVF )1/2 . The following correspondence is obvious, compare (2.1): Remark 2.25. Let A be a self-adjoint and V a symmetric operator in a Krein space K , [·, ·] such that A and V are bounded from below in K , [·, ·] and D (a) ⊂ D (v), where a and v are the forms associated with A and VF , respectively. Then V is relatively form-bounded with respect to A and relative formbound β0 in the Krein space K ,[·, ·] if and only if JV is relatively formbounded with respect to J A and relative form-bound β0 in the Hilbert space K , (·, ·) in the usual sense. As in the case of relatively bounded operators, the constant α in equation (2.3) may have to be chosen very large if β is chosen very close to the relative form-bound β0 . Remark 2.26. Note that if V is relatively form-bounded with respect to A with relative form-bound zero, then for any β > 0 there exists an αβ > 0 such that (2.4)

|v x| ≤ αβ x 2 + β|a x|,

x ∈ D (a).

The following theorem shows that relative form-boundedness is weaker than relative boundedness, see [Kat95, Theorem VI.1.38]. operator in a Krein Theorem 2.27. Let A be a self-adjoint and V a symmetric space K , [·, ·] such that A is bounded from below in K , [·, ·] . Suppose that V is relatively bounded with respect to A and A-bound β0 . Then V is relatively form-bounded with respect to A with relative form-bound ≤ β0 . In particular, if V has A-bound 0, then also the relative form-bound of V with respect to A is 0.

44


Proof. According to Remark 1.17 we have JV x ≤ α x + β J Ax ,

x ∈ D (A),

where α ≥ 0 and β ≥ β0 ≥ 0. A is bounded from below in K , [·, ·] and thus J A is bounded from below in K , (·, ·) . Hence, by [Kat95, Theorem V.4.11], the operator J A ε , where A ε := A + εV , ε ∈ R, is self-adjoint and bounded from below in the Hilbert space K , (·, ·) if |ε| < β−1 and the lower bound γ J A ε of J A ε satisfies α γ J A ε ≥ γ J A − |ε| max , α + β|γ J A | , 1 − β|ε|

where γ J A is the lower bound of J A. Hence (2.5)

−ε[V x, x] = −ε(JV x, x) ≤ −γ J A ε x 2 + (J Ax, x) = −γ J A ε x 2 + [Ax, x],

for x ∈ D (A). Case (i): [V x, x] ≥ 0. Since ε can be chosen arbitrarily close to −β−1 and thus to −β0−1 , we obtain from (2.5) with ε0 := −ε > 0 −γ J A ε 1 x 2 + [Ax, x], x ∈ D (A). [V x, x] = [V x, x] ≤ max 0, ε0 ε0 Case (ii): [V x, x] < 0. Since ε can be chosen arbitrarily close to β−1 and thus to β0−1 , we obtain from (2.5) with ε0 := ε > 0 −γ J A ε 1 −[V x, x] = [V x, x] ≤ max 0, x 2 + [Ax, x], x ∈ D (A). ε0 ε0 Altogether, the form [V x, x] is relatively bounded with respect to the form [Ax, x] with relative form-bound ≤ β0 . By Remark 2.22, the same is true for their closures, that is, V is relatively form-bounded with respect to A with ■ relative form-bound less than or equal to β0 . Definition and Remark 2.28. Let A be a self-adjoint operator in a Krein space K , [·, ·] such that A is non-negative in K , [·, ·] . Set (x, y)a := (x, y) + a x, y,

x, y ∈ D (a),

where a is the form associated with A. Then (·, ·)a defines a positive defi nite inner product on D (a) and D (a), (·, ·)a becomes a Hilbert space, which we denote by Da (compare Definition and Remark 1.14). The norm which is induced by (·, ·)a is denoted by · a and (·, ·)a is called graph inner product.


45

Remark 2.29. Let A be a self-adjoint and V a symmetric operator in a Krein space K , [·, ·] such that A and V are non-negative in K , [·, ·] and D (a) ⊂ D (v), where a and v are the forms associated with A and VF , respectively. Suppose that V is relatively form-bounded with respect to A. Then the )1/2 := (JV )1/2 restriction (JV as an operator from Da to K is bounded. D (a)

Proof. By assumption, inequality (2.3) holds. Define μ := max{α, β}. Then we have

2

1/2 2

) x = (JV )1/2 x = |v x| ≤ μ x 2 + μ|a x| = μ x 2a , x ∈ D (a), (2.6) (JV )1/2 is bounded. i.e., (JV )1/2 is bounded from Da to K , then Vice versa, if (JV

1/2 2 1/2 2

(JV ) x ≤ (JV ) x 2a = α x 2 + βa x|,

)1/2 2 . where α := β := (JV

x ∈ D (a), ■

As in Chapter 1, we are particularly interested in relatively form-bounded operators with relative form-bound 0. A sufficient condition is given by Lemma 2.31 below. In the following definition of relatively form-compact operators we assume for simplicity that A and V are non-negative in a Krein space; a more general definition can be found, e.g., in [GMMN09, Definition 3.1]. Definition 2.30. Let A be a self-adjoint and V a symmetric operator in a Krein space K , [·, ·] such that A and V are non-negative in K , [·, ·] . Let a and v be the forms associated with A and VF , respectively. Let V be relatively form-bounded with respect to A in the Krein space K , [·, ·] . Then V is called relatively form-compact with respect to A if and only if the restriction 1/2 (JV ) D (a) as an operator from Da to K is a compact map. The following theorem is the analogue of Theorem 1.22 for relatively form-compact operators. Lemma 2.31. Let A be a self-adjoint and V a symmetric operator in a Krein space K , [·, ·] such that A and V are non-negative in K ,[·, ·] . If V is relatively form-compact with respect to A, then the relative form-bound of V with respect to A is 0.

46


Proof. The proof is similar to the proof of Theorem 1.22. We prove the theV is not 0. orem by contradiction. Suppose that the relative form-bound of 1/2 Then there exist an ε > 0 and a sequence (xn )∞ ⊂ D (J A) such that for n=1 all natural numbers n (2.7)

(JV )1/2 xn 2 > n xn 2 + ε (J A)1/2 xn 2 .

Set yn := xn / xn a . Then, by inequalities (2.3), (2.6) and (2.7), and since yn a = 1, n ∈ N, we have for n ≥ ε

(2.8)

2

max{α, β} = max{α, β} yn 2a ≥ (JV )1/2 yn

2 > n yn 2 + ε (J A)1/2 yn

2

≥ ε yn 2 + (J A)1/2 yn ,

where α and β are constants according to (2.3). Hence, by inequality (2.8), ∞ 1/2 yn → 0 if n → ∞ and (yn )∞ and (J A) y n n=1 are bounded sequences. n=1 with respect to A, there exists a subThus, since V is relatively form-compact 1/2 ∞ ∞ sequence (yn k )∞ of (y ) such that V y converges to some z ∈ K . n n k k =1 n=1 k=1 1/2 ∞ Since V be closable, (yn k )∞ converges to zero and V y k=1 converges k=1 1/2 n k ∞ to z, it follows that z = 0. By inequality (2.8), also A yn k k=1 converges to 0. This leads to the contradiction 1 = yn k 2a = yn k 2 + A 1/2 yn k 2 −→ 0,

k → 0.

■

2.2 Continuity of Separated Parts of the Spectrum If V is relatively form-bounded with respect to A 0 with A 0 -form-bound less than 1, then the spectrum of A 0 + εV , ε ∈ [0, 1], cannot suddenly expand for increasing ε. As in Chapter 1, we are interested in the case when the spectrum σ(A 0 ) consists only of isolated eigenvalues or isolated parts. In the first paragraph we present some well-known perturbation results for self-adjoint operators in a Hilbert space. In the second paragraph of this section we extend the obtained results to the case of self-adjoint operators in Krein spaces. Analogously to Chapter 1, if the spectrum σ(A 0 ) is real, then our main results state conditions which guarantee that also the spectrum of A 0 V is real, even when A 0 V is not self-adjoint in a Hilbert space.


47

2.2.1 Perturbation of Spectra of Self-Adjoint Operators in Hilbert Spaces In this paragraph we consider the case where the spectrum σ(A 0 ) of a selfadjoint operator A 0 in a Hilbert space H contains a bounded subset σ1 (A 0 ) separated from the rest σ(A 0 )\σ1 (A 0 ) by a rectifiable closed curve Γ. For symmetric perturbations V which are relatively form-bounded with respect to A 0 , we state conditions for the spectrum A 0 V to be also separated into two parts by Γ. We recall the following theorem, which was proved in [Kat95, Theorem VI.3.9]. Theorem 2.32 (Cf. [Kat95, Theorem VI.3.9 and Remark VI.3.10]). In a Hilbert space H let a0 be a densely defined, closed symmetric form bounded from below with associated self-adjoint operator A 0 . Let v be a form relatively bounded with respect to a0 so that D (v) ⊃ D (a0 ) and |v x| ≤ α x 2 + βa0 x,

x ∈ D (a0 ),

where 0 ≤ β < 1 but α may be positive, negative or zero. Then a1 = a0 + v is sectorial and closed. The associated operator A 1 is m-sectorial; A 1 is selfadjoint if v is symmetric. Let κ = 1 if v symmetric, otherwise κ = 2. If z ∈ ρ (A 0 ) and

κ (α + β A 0 )(A 0 − z)−1 < 1,

(2.9) then z ∈ ρ (A 1 ) and (2.10)

4κ (α + β A 0 )(A 0 − z)−1

(A 0 − z)−1

(A 1 − z)−1 − (A 0 − z)−1 < .

2 1 − κ (α + β A 0 )(A 0 − z)−1

For symmetric v, results similar to Theorem 2.32, which also show that A 1 is self-adjoint, can be found in [Sim71b, Theorem II.7] (KLMN1 theorem) or [Nel64, Appendix], but they do not include inequality (2.10). In the following we apply Theorem 2.32 to families of operators A ε = A 0 εV , ε ∈ [0, 1], to establish convergence in the generalized sense for ε → 0. 1 KLMN stands for Kato, Lions, Lax, Milgram and Nelson, compare [Sim71a].

48


Remark 2.33. If in Theorem 2.32, instead of a1 , the sum of forms aε := a0 + εv, ε ∈ [0,1], is considered, then inequality (2.10) becomes (see the proof of [Kat95, Theorem VI.3.9] or Theorem 2.39 below)

4κ (α + β A 0 )(A 0 − z)−1

(A 0 − z)−1

(A ε − z)−1 − (A 0 − z)−1 < ε .

2 1 − κ (α + β A 0 )(A 0 − z)−1 Hence

(A ε − z)−1 − (A 0 − z)−1 −→ 0,

ε → 0,

and, by Theorem 1.27, A ε converges to A 0 in the generalized sense. Corollary 2.34. Let A 0 be a self-adjoint and V a symmetric operator in a Hilbert space H such that A 0 and V are bounded from below in H . Further, let V be relatively form-bounded with respect to A 0 such that (2.3) holds with constants α ∈ R and β ∈ [0, 1). Define the self-adjoint family of operators A ε , ε ∈ [0, 1], as the operator associated with the sum of forms aε := a0 + εv, ε ∈ [0, 1], with a0 and v being the forms associated with A 0 and VF , respectively. Suppose that the spectrum of A 0 is separated into two parts σ1 (A 0 ) and σ2 (A 0 ) by a Cauchy contour Γ. If

(2.11) sup (α + β A 0 )(A 0 − z)−1 < 1, z∈Γ

then the spectrum of A 1 = A 0 V is likewise separated into two parts σ1 (A 1 ) and σ2 (A 1 ), Γ ⊂ ρ (A 1 ) and the results of Theorem 1.35 hold. Proof. The assumptions of Theorem 2.32 are satisfied. Hence Γ⊂ρ (A 1 ). Define the isolated part σ1 (A 1 ) := int Γ ∩ σ(A 1 ) of σ(A 1 ) and σ2 (A 1 ) := σ(A 1 )\σ1 (A 1 ). By Lemma 1.33, A 1 can be decomposed according to H = M1(A 1 ) ⊕ M2 (A 1 ). We have M1 = E(A 1 , σ1 (A 1 ))H and M2 = 1 − E(A 1 , σ1 (A 1 )) H . According to Remark 2.33, (A ε − z)−1 , z ∈ Γ, depends continuously on ε for 0 ≤ ε ≤ 1. Hence the Riesz projection E(A ε , σ1 (A ε )) of A ε corresponding to σ1 (A ε ) is continuous in ε for 0 ≤ ε ≤ 1. The last part of the proof is analogous to that of Theorem 1.35. ■ Theorem 2.32 and Corollary 2.34 provide the necessary tools to obtain results analogous to those for relatively bounded operators in the first chapter, see Theorems 1.39 and 1.40. Again we distinguish the following situations: (a) We consider one isolated eigenvalue (an infinite sequence of isolated eigenvalues, respectively) of the unperturbed operator A 0 .


49

(b) We consider one isolated compact part (an infinite sequence of isolated compact parts, respectively) of the spectrum of the unperturbed operator A 0 . Note that in the theorems below, in contrast to Theorems 1.39 and 1.40, the operator A 1 is self-adjoint in the Hilbert space by Theorem 2.32 since V is symmetric in the Hilbert space. In situation (a) we obtain the following result. Theorem 2.35. Let A 0 be a self-adjoint and V a symmetric operator in a Hilbert space H such that A 0 and V are bounded from below in H . Further, let V be relatively form-bounded with respect to A 0 with relative form-bound less than 1/2 and let A ε , ε ∈ [0, 1], be the family of self-adjoint operators in H which is associated to the sum of forms aε := a0 + εv, ε ∈ [0, 1], with a0 and v being the forms associated to A 0 and VF , respectively. (i) Let λ0 ∈ R be an isolated eigenvalue of A 0 with multiplicity m < ∞ and set δ := dist λ0 , σ(A 0 )\{λ0 } . If (2.3) holds with constants α ≥ 0 and β ∈ [0, 1/2) such that 1 1 α + β δ + |λ0 | < , (2.12) δ 2 0 0 then σ1 := σ(A 1 ) ∩ λ − δ/2, λ + δ/2 consists of a finite system of isolated (real) eigenvalues with total multiplicity m. 0 0 (ii) Let A 0 have discrete spectrum consisting of eigenvalues 0 λ1 < λ2 0 (otherwise consider the operators − A 0 and −V ). Let z = x + iy, such that x ∈ (a, b) and y ∈ R. That is, z ∈ (a, b) + iR ⊂ ρ (A 0 ). Then, according to Theorem 2.32, z ∈ ρ (A 1 ) if

(α + β A 0 )(A 0 − z)−1 < 1. (2.17)

52


Since A 0 is self-adjoint in the Hilbert space H , it is possible to apply Proposition 1.38. Thus the fact that |λ − x| ≤ |λ − z| for λ ∈ σ(A 0 ) ⊂ R\(a, b) implies that condition (2.17) is satisfied if (2.18)

sup

α + β|λ|

< 1.

|λ − x |

λ∈σ(A 0 )

Define the function f : C → R by f (λ) :=

α + β|λ|

|λ − x |

.

Now the supremum in (2.18) can be estimated (see [Ves08, Theorem 3.1]). We distinguish the two cases a > 0 and a ≤ 0. Case (i): a > 0. Since x ∈ (a, b) also x > 0. If λ ≤ 0, then (2.19)

d d α − βλ −β(x − λ) + (α − βλ) α − βx f (λ) = = = 2 dλ dλ x − λ (x − λ) (x − λ)2

and hence sup f (λ) =

(2.20)

λ≤0

⎧ ⎪ ⎨β, α ⎪ ⎩ , x

if α ≤ β x, if β x < α.

If λ ∈ [0, a], then (2.21)

d d α + βλ β(x − λ) + (α + βλ) α + βx f (λ) = = = dλ dλ x − λ (x − λ)2 (x − λ)2

and hence sup f (λ) = max f (λ) =

(2.22)

0≤λ≤a

0≤λ≤a

α + βa

x−a

>

α

x

.

If λ ≥ b, then (2.23)

d d α + βλ β(λ − x) − (α + βλ) −α − β x f (λ) = = = dλ dλ λ − x (λ − x)2 (λ − x)2

and hence (2.24)

sup f (λ) = max f (λ) = b≤λ

b≤λ

α + βb

b−x

> β.


53

Formulas (2.20), (2.22) and (2.24) now imply

sup f (λ) = max

λ∉(a,b)

α + βa α + β b , x−a b−x

which is less than 1 if x ∈ I. Case (ii): a ≤ 0. If λ ≤ a, then, by equation (2.19),

(2.25)

sup f (λ) = λ≤a

⎧ ⎪ ⎨β,

if α ≤ β x,

⎪ ⎩ α − βa , x−a

if β x < α.

If λ ≥ b, then, by equation (2.23),

(2.26)

sup f (λ) = λ≥b

⎧ ⎪ ⎨β,

if α + β x ≤ 0,

⎪ ⎩ α + βb , b−x

if 0 < α + β x.

In this case equations (2.25) and (2.26) imply

sup f (λ) = max

λ∉(a,b)

α − βa α + β b , x−a b−x

which is less than 1 if x ∈ I.

■

2.2.2 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces In this paragraph we extend the results of the previous section to operators which are self-adjoint in Krein spaces. In the following we generalize Theorem 2.32 proved in [Kat95, Theorem VI.3.9] to the case when the operator A 1 is no longer sectorial. This is needed in order to treat perturbations V which are semi-bounded in a Krein space. in a Theorem 2.39. Let A 0 be a self-adjoint and V a symmetric operator Krein space K ,[·, ·] such that A 0 and V are bounded K from below in , [ ·, ·] and A 0 is also self-adjoint in the Hilbert space K , (·, ·) . Further, let V be relatively form-bounded with respect to A 0 such that (2.3) holds with constants α ≥ 0 and β ∈ [0,1). Then the operator A 1 which corresponds to the sum of

54


forms a1 := a0 + v, where a0 and v are the forms associated with A 0 and VF , respectively, is self-adjoint. If z ∈ ρ (A 0 ) and

(2.27) α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1, then z ∈ ρ (A 1 ) and

4 α (A 0 − z)−1 +β A 0 (A 0 − z)−1 (A 0 − z)−1

− 1 − 1 (2.28) (A 1 − z) −(A 0 − z) < .

2 1 − α (A 0 − z)−1 + β A 0 (A 0 − z)−1

0 := J A 0 and V := JV Proof. Since A 0 = A [0∗] and V ⊂ V [∗] the operators A are self-adjoint and symmetric, respectively, in the Hilbert space K , (·, ·) . Thus the self-adjointness of A 1 follows immediately from Theorem 2.32.

The proof of inequality (2.28) is similar to the proof of [Kat95, Theorem VI.3.9]. Assume that b > 0 and set a0 = a0 + αβ−1 + δ and a1 = a1 + αβ−1 + δ for some δ > 0 which will be determined later.Here a0 and a1 are the forms The opassociated with A 0 and A 1 in the Krein space K , [·, ·] , respectively. erators associated with a0 and a1 in the Krein space K ,[·, ·] are respectively A 0 = A 0 + αβ−1 J + δ J and A 1 = A 1 + αβ−1 J + δ J by equation (2.1). Inequality (2.3) implies a0 + αβ−1 ≥ 0 and thus a0 ≥ δ in K , [·, ·] . By (2.3), we also have |v x| ≤ βa0 x. According to [Kat95, Lemma VI.3.1], v x, y may be written in the form (CG x,G y), x, y ∈ D (G ), where C is a linear operator with C ≤ β 1/2 and G := J A 0 . Since J A 0 ≥ δ in K , (·, ·) we have G ≥ δ1/2 in K , (·, ·) and hence G −1 ≤ δ−1/2 . Furthermore a0 x, y = [A 0 x, y] = (J A 0 x, y) = (J A 0 )1/2 x, (J A 0 )1/2 y = G x,G y and thus

a1 x, y = (a0 + v) x, y = (1 + C)G x,G y .

Consequently, noting that G is self-adjoint in the Hilbert space K ,(·, ·) ,

J A 1 + αβ−1 + δ = J A 1 = G (1 + C)G . Since, by assumption, A 0 = A [0∗] and also A 0 = A ∗0 , we have J A 0 = (J A 0 )∗ = A ∗0 J = A 0 J, that is, J commutes with A 0 . Hence J commutes with J A 0 and thus also with (J A 0 )1/2 and (J A 0 )−1/2 , see [GGK90, Section V.4 ff.]. Let z ∈ ρ (A 0 ). We obtain A 1 − z = JG 1 − αβ−1 + δ (J A 0 )−1 − z(J A 0 )−1/2 J(J A 0 )−1/2 + C G = JG 1 − αβ−1 + δ + zJ (J A 0 )−1 + C G .


55

Thus (2.29)

−1 (A 1 − z)−1 = G −1 1 − αβ−1 + δ + zJ (J A 0 )−1 + C G −1 J,

provided the second factor on the right exists and is bounded. This will be shown as follows. According to [Kat95, I-(4.24)] (the Neumann series), the factor −1 1 − αβ−1 + δ + zJ (J A 0 )−1 + C −1 exists and is bounded if 1 − αβ−1 + δ + zJ (J A 0 )−1 exists and has norm less than C −1 . Since z ∈ ρ (A 0 ), the operator A 0 − z is bijective, and hence also J A 0 − zJ. Consequently, 1 − αβ−1 + δ + zJ (J A 0 )−1 = J A 0 − αβ−1 + δ + zJ (J A 0 )−1 = J A 0 + αβ−1 J + δ J − αβ−1 + δ + zJ (J A 0 )−1 = J A 0 − zJ (J A 0 )−1 is bijective. Since C ≤ β, it remains to show that β M < 1, where

−1

−1 −1

. M = 1 − αβ + δ + zJ (J A ) 0

For M we have the estimate

−1

−1 −1

M = 1 − αβ + δ + zJ (J A ) 0

−1 −1

= J A J A − αβ + δ + zJ 0 0

(2.30)

−1 −1 −1 −1

= J A 0 + αβ + δ J A 0 + αβ + δ − αβ + δ + zJ

−1 −1

=

β α + β J A 0 + δ J(A 0 − z)

≤ β−1 α + β J A 0 (A 0 − z)−1 + δ (A 0 − z)−1

≤ β−1 α (A 0 − z)−1 + β A 0 (A 0 − z)−1 + δ (A 0 − z)−1 .

Now let δ = a(1 − a)(1 + a)−1 b−1 where

a = α (A 0 − z)−1 + β A 0 (A 0 − z)−1

and

b = β (A 0 − z)−1 .

56


Since a < 1 by assumption (2.27), we have δ > 0. Then, by (2.30), we have M ≤ β−1 a + δβ−1 b and thus β M < 1 since β M ≤ a + δb = a +

a(1 − a) 2a = < 1, 1+ a 1+a

which is guaranteed by (2.27). Thus, −1 according to [Kat95, I-(4.24)], the factor 1 − αβ−1 + δ + zJ (J A 0 )−1 + C in (2.29) exists and is bounded.

A similar expression for (A 0 − z)−1 is obtained by setting the linear operator C = 0 in (2.29): (2.31)

−1 (A 0 − z)−1 = G −1 1 − αβ−1 + δ + zJ (J A 0 )−1 G −1 J.

By [Kat95, I-(4.24)],

−1 −1 2

−1 −1

1 − αβ−1 + δ + zJ + C (J A )−1

≤ βM , − 1 − αβ + δ + zJ (J A )

1 − βM 0 0

for β M < 1, and hence inequalities (2.29) and (2.31) yield

(A 1 − z)−1 − (A 0 − z)−1 ≤

(2.32) Since

βM2 =

and

βM 2

(1 − β M)δ

.

(β M)2 4a2 ≤ β β(1 + a)2

2a a(1 − a) a(1 − a)2 = , (1 − β M)δ ≥ 1 − 1 + a b(1 + a) b(1 + a)2

inequality (2.28) follows from (2.32) and

4a (A 0 − z)−1 4a2 b(1 + a)2 4ab ≤ · = = . (1 − β M)δ β(1 + a)2 a(1 − a)2 β(1 − a)2 (1 − a)2 βM2

The case β = 0 can be dealt with by going to the limit β → 0.

■

Remark 2.40. In contrast to Theorem 2.32 it is not possible to choose α < 0 in (2.27) of Theorem 2.39. This is due to inequality (2.30) which requires the stronger condition (2.27) compared to inequality (2.9) for the Hilbert space case.


57

Remark 2.41. Consider in Theorem 2.39 instead of A 1 the family of operators A ε , ε ∈ [0,1], which correspond to the sum of forms aε := a0 + εv, ε ∈ [0, 1], with a0 and v being the forms associated with A 0 and VF , respectively. Then inequality (2.27) becomes (see proof of Theorem 2.39 above)

4 α (A 0 − z)−1 + β A 0 (A 0 − z)−1 (A 0 − z)−1

(A ε − z)−1 − (A 0 − z)−1 < ε .

2 − 1 − 1

+ β A 0 (A 0 − z) 1 − α (A 0 − z) Hence

(A ε − z)−1 − (A 0 − z)−1 −→ 0,

ε → 0,

and, by Theorem 1.27, A ε converges to A 0 in the generalized sense. 2.42. Let A 0 , V and A 1 be linear operators in a Krein space Corollary K , [·, ·] as in Theorem 2.39. Suppose that the spectrum of A 0 is separated into two parts σ1 (A 0 ) and σ2 (A 0 ) by a Cauchy contour Γ. If

(2.33) sup α (A 0 − z)−1 + β A 0 (A 0 − z)−1 < 1, z ∈Γ

then the spectrum of A 1 is likewise separated into two parts σ1 (A 1 ) and σ2 (A 1 ), Γ ⊂ ρ (A 1 ) and the results of Theorem 1.35 hold. Proof. With Theorem 2.39 instead of Theorem 2.32 the proof is analogous to ■ the proof of Corollary 2.34. The preceding results and Theorem 1.43 enable us to formulate Krein space versions of Theorems 2.35 and 2.36. in a Theorem 2.43. Let A 0 be a self-adjoint and V a symmetric operator Krein space K ,[·, ·] such that A 0 and V are bounded from below in K , [·, ·] and A 0 is also self-adjoint in the Hilbert space K ,(·, ·) . Further, let V be relatively form-bounded with respect to A 0 with relative form-bounded less than 1/2 and let A ε , ε ∈ [0, 1], be the family of operators which correspond to the sum of forms aε := a0 + εv, ε ∈ [0,1], with a0 and v being the forms associated with A 0 and VF , respectively. (a) Claims (i) and (ii) of Theorem 2.35 hold, respectively, if (i) the eigenvalue λ0 is of definite type; in this case the eigenvalues contained in σ1 are of the same type as λ0 .

58


(ii) the eigenvalues λ0n , n ∈ N, are of definite type; in this case the eigenvalues contained in σn,ε are of the same type as λ0n for n ∈ N and ε ∈ [0, ε0 ]. (b) Claims (i) and (ii) of Theorem 2.36 hold, respectively, if (i) the spectral subspace E(A 0 , σ0 )K is uniformly definite; in this case E(A ε , σε )K is of the same type as E(A 0 , σ0 )K for ε ∈ [0, 1]. (ii) the spectral subspace E(A 0 , σn,0 )K , n ∈ N, is uniformly definite; in this case E(A ε , σn,ε )K is of the same type as E(A 0 , σn,0 )K for n ∈ N and ε ∈ [0, ε0 ]. Proof. We prove claim (i) of (a). The first part of the proof is analogous to the proof of Theorem 2.35. As in the proof of Theorem 2.35, let Γ be the positively λ0 and radius δ/2. Then Γ ⊂ ρ (A 0 ), oriented curve along the circle with center 0 0 {λ } ⊂ int (Γ) and Γ ∪ int (Γ) ∩ σ(A 0 )\{λ } = .By Proposition 1.38, since A 0 is also self-adjoint in the Hilbert space K , (·, ·) , inequality (2.33) is satisfied if (2.12) holds. By Corollary 2.42, the spectrum of A ε is separated into the two parts σε := int Γ ∩ σ(A ε ) and σ(A ε )\σε such that σε consists of isolated eigenvalues with total multiplicity m. By Theorem 2.39 and Remark 2.41, for one (and hence for all) z ∈ ρ (A ε ) the resolvents of A ε = A 0 + εV , depend apply continuously on ε in the operator norm for 0 ≤ ε ≤ 1. Let n ∈ N. We Theorem 1.43. Since A ε is self-adjoint in the Krein space K ,[·, ·] , σ(A ε ) is symmetric with respect to the real axis. Hence σε = σ∗ε for all 0 ≤ ε ≤ 1 since Γ = Γ∗ . Since, by assumption, λ0 is of definite type, E(A 0 , σ0 )K is either uniformly positive or uniformly negative. According to Theorem 1.43 the subspace E(A ε , σε )K is of the same type as E(A 0 , σ0 )K and the set σε is real for all 0 ≤ ε ≤ 1. Consequently, σε consists of a finite system of real eigenvalues of A ε with total multiplicity m which are of the same type as λ0 for 0 ≤ ε ≤ 1. Claim (ii) of (a) follows from (i) since the assumptions of (i) are satisfied for A 0 and εV with ε ∈ [0, ε0 ] for every eigenvalue λ0n , n ∈ N, if we choose ε0 such that 1 1 0 ε0 sup αn + βn δn + |λ n | < . δ 2 n∈N n

The proof of claims (i) and (ii) of (b) are analogous to those of (i) and (ii) of (a). ■

2.3 Pseudo-Friedrichs Extensions

59

2.3 Pseudo-Friedrichs Extensions While the Friedrichs extension is defined for densely defined sectorial operators, this section introduces another kind of extension of the operatorsum A 0 + V , which can be applied to not necessarily sectorial operators. In contrast to the preceding paragraphs the main difference is the assumption D (V ) ⊂ D (A 0 ). The results of this section are not essentially related to sesquilinear forms, but the techniques used in the proofs are similar.

2.3.1 Perturbation of Spectra of Self-Adjoint Operators in Krein Spaces The following theorem is a Krein space-generalisation of [Kat95, Theorem VI.3.11]. This result has been proved in a different way in [Ves72a], see Lemma 2.2 therein. Theorem 2.44. Let A 0 be a operator in a Krein space K , [·, ·] self-adjoint and let V be an operator in K ,[·, ·] such that D (V ) ⊂ D (A 0 ) and

(2.34)

|[V x, x]| ≤ α x 2 + β [J | A 0 | x, x],

x ∈ D (V ),

where α ≥ 0 and 0 ≤ β < 1 or 0 ≤ β < 1/2 according to whether V is symmetric in K , [·, ·] or not. J denotes a fundamental symmetry on K . If D (V ) is a core there exists a unique closedextension A 1 of A 0 + V such that of | A 0 |1/2 , then [∗] 1/2 D (A 1 ) ⊂ D | A 0 |1/2 and D (A . A 1 is self-adjoint in K , [·, ·] if 1 ) ⊂ D | A0| V is symmetric in K , [·, ·] . Definition 2.45. The operator A 1 from Theorem 2.44 is called pseudoFriedrichs extension of A 0 + V in the Krein space K ,[·, ·] . Proof of Theorem 2.44. The proof follows from the Hilbert space version of the theorem, see [Kat95, Theorem VI.3.11]. Let (·, ·) denote a Hilbert space inner product on K , i. e., (·, ·) = [J ·, ·]. Then inequality (2.34) yields |(JV x, x)| = |[V x, x]| ≤ α x 2 + β[J | A 0 | x, x] = α x 2 + β(| A 0 | x, x),

x ∈ D (V ).

By assumption, A 0 = A [0∗] or, equivalently, J A 0 = (J A 0 )∗ . Thus 1/2 | A 0 | = (A ∗0 A 0 )1/2 = (A ∗0 J 2 A 0 )1/2 = (J A 0 )∗ (J A 0 ) = | J A 0 |.

60


Since | A 0 | = | J A 0 | and the operators A 0 and J A 0 as well as V and JV have the same domains, respectively, [Kat95, Theorem VI.3.11] implies the exis1 of J A 0 + JV . That is, tence of the unique pseudo-Friedrichs extension A (2.35)

1 ) ⊃ D (J A 0 + JV ) D(A

and

1 x = (J A 0 + JV )x, A

1 has domain D (A 1 ) = for x ∈ D (J A 0 + JV ) = D (V ). The operator A 1 := J A D (J A 1 ) = D ( A 1 ) and since D (J A 0 + JV ) = D (A 0 + V ), (2.35) implies D (A 1 ) ⊃ D (A 0 + V ) and 1 x = J(J A 0 + JV )x = (A 0 + V )x, A1 = J A

x ∈ D (A 0 + V ).

That is, A 1 is the (unique) pseudo-Friedrichs extension of A 0 + V in the Krein space K . If V is symmetric with respect to [·, ·], then, by Lemma 1.8, JV is symmetric with respect to (·, ·). In this case, according to [Kat95, Theo1 is self-adjoint with respect to (·, ·), and hence A 1 is selfrem VI.3.11], A adjoint with respect to the Krein space inner product [·, ·] by Lemma 1.8. ■ Although the following result is rather convenient, it has has not explicitly been stated but proved in [Kat95]. It is a corollary to [Kat95, Theorem VI.3.11]. Lemma 2.46. Let A 0 be a self-adjoint operator in a Hilbert space H , (·, ·) and let V be an operator such that D (V ) ⊂ D (A 0 ) and |(V x, x)| ≤ α x 2 + β (| A 0 | x, x),

x ∈ D (V ),

where α ≥ 0 and 0 ≤ β < 1 or 0 ≤ β < 1/2 according to whether V is symmetric in H or not. Let D (V ) be a core of | A 0 |1/2 and denote the pseudo-Friedrichs extension of A 0 + V by A 1 . Let κ = 1 if V is symmetric in H ,(·, ·) , otherwise κ = 2. If there is a point z ∈ ρ (A 0 ) such that

κ α + β| A 0 | (A 0 − z)−1 < 1,

(2.36) then z ∈ ρ (A 1 ) and

4κ α + β| A 0 | (A 0 − z)−1

(A 0 − z)−1

(A 1 − z)−1 − (A 0 − z)−1 < .

2

− 1 1 − κ α + β| A 0 | (A 0 − z)


61

Proof. The claim follows from the proof of [Kat95, Theorem VI.3.11], compare ■ inequality (3.24) therein. The following theorem gives a convenient criterion for spectral gaps between separated parts of the spectrum not to close; it has also been proved in [Ves08, Theorem 3.1]. Theorem 2.47. Let A 0 be a self-adjoint and V a symmetric operator in a Hilbert space H ,(·, ·) such that D (V ) ⊂ D (A 0 ) and |(V x, x)| ≤ α x 2 + β (| A 0 | x, x),

x ∈ D (V ),

where α ≥ 0 and 0 ≤ β < 1. Let D (V ) be a core of | A 0 |1/2 and denote the pseudoFriedrichs extension of A 0 + V by A 1 . Suppose the open interval (a, b) is a subset of ρ (A 0 ). Define the interval I by I : = a + α + β| a | , b − α + β| b | . If

| a| + | b | 1 1 α+β < , δ 2 2

where δ = b − a, then I = and I is a subset of ρ (A 1 ). Proof. The proof is analogous to that of Theorem 2.38.

■

It is possible to formulate a Krein space version of Lemma 2.46: Theorem 2.48. Let A 0 and V be linear operators in a Krein space K , [·, ·] as in Theorem 2.44. Further, let A 0 be self-adjoint in the Hilbert space K ,(·, ·) . Denote the pseudo-Friedrichs extension of A 0 + V by A 1 . Let κ = 1 if V is , [ · , · ] , otherwise κ = 2. If there is a point z ∈ ρ (A 0 ) such that symmetric in K

(2.37) κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1 < 1,

then z ∈ ρ (A 1 ) and

(A 1 − z)−1 − (A 0 − z)−1

4κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1 (A 0 − z)−1 (2.38) < .

2 − 1 − 1

1 − κ α (A 0 − z) + β | A 0 |(A 0 − z)

62


Proof. From [Kat95, VI-(3.21)] we obtain the identity J A 1 = G J A 0 (J A 0 )−1 + C G ,

where G := (J A 0 )1/2 , A 0 = | A 0 |+ αβ−1 J + δ J with δ > 0 to be determined later and C is a bounded linear operator in K ,[·, ·] with C ≤ κβ; note that since A 0 = A [0∗] or, equivalently, J A 0 = (J A 0 )∗ , we have | J A 0 | = | A 0 |. Since, further, A 0 = A ∗0 , we have J A 0 = (J A 0 )∗ = A ∗0 J = A 0 J, that is, J commutes with A 0 . Hence J commutes with J A 0 and thus also with (J A 0 )1/2 and (J A 0 )−1/2 , see [GGK90, Section V.4 ff.]. Let z ∈ ρ (A 0 ). Then A 1 − z = JG J A 0 (J A 0 )−1 − z(J A 0 )−1/2 J(J A 0 )−1/2 + C G = JG (J A 0 − zJ)(J A 0 )−1 + C G ,

and thus (2.39)

−1 (A 1 − z)−1 = G −1 J(A 0 − z)(J A 0 )−1 + C G −1 J,

provided the second factor on the right exists and is bounded. According to [Kat95, I-(4.24)] (the Neumann series), this is true if

−1

J(A 0 − z)(J A )−1 < 1 ,

C 0

(2.40)

which is satisfied if

−1

−1

C J(A − z)(J A ) 0 0

≤ C J A 0 (A 0 − z)−1

≤ κβ J | A 0 | + αβ−1 + δ (A 0 − z)−1 (2.41)

≤ κ α + β J | A 0 | (A 0 − z)−1 + κβδ (A 0 − z)−1

≤ κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1 + κβδ (A 0 − z)−1 < 1.

Now let δ = a(1 − a)(1 + a)−1 b−1 where

a = κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1

and

b = κβ (A 0 − z)−1 .


63

By assumption (2.37), a < 1 and thus δ > 0. Then, by (2.41) and since

κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1 + κβδ (A 0 − z)−1 (2.42) a(1 − a) 2a = = a+ , 1+a 1+a inequality (2.40) is satisfied since a < 1 by (2.37). Equation (2.39) and a similar expression for (A 0 − z)−1 obtained by setting the linear operator C = 0 in (2.39) yield, by [Kat95, I-(4.24)],

2

−1 κβ J A (A − z)

0

0

(A 1 − z)−1 − (A 0 − z)−1 ≤

.

1 − κβ J A 0 (A 0 − z)−1 δ

(2.43)

Since, by (2.41) and (2.42),

2

2 κβ J A 0 (A 0 − z)−1 4a2

−1 κβ J A 0 (A 0 − z) = ≤ κβ κβ(1 + a)2

and

2a a(1 − a) a(1 − a)2

1 − κβ J A 0 (A 0 − z)−1 δ ≥ 1 − = , 1 + a b(1 + a) b(1 + a)2

inequality (2.38) follows from (2.43) and

2

κβ J A 0 (A 0 − z)−1 4a (A 0 − z)−1 .

≤ (1 − a)2

− 1 1 − κβ J A 0 (A 0 − z) δ

The case β = 0 can be dealt with by going to the limit β → 0.

■

Remark 2.49. Consider in Theorem 2.48 instead of A 1 the family of pseudoFriedrichs extensions A ε , ε ∈ [0,1], of A 0 + εV . Analogously to Remark 2.41, inequality (2.38) implies that A ε converges to A 0 in the generalized sense. Corollary 2.50. Let A 0 , V and A 1 be linear operators in a Krein space K , [·, ·] as in Theorem 2.48. Suppose that the spectrum of A 0 is separated into two parts σ1 (A 0 ) and σ2 (A 0 ) by a Cauchy contour Γ. If

κ α (A 0 − z)−1 + β | A 0 |(A 0 − z)−1 < 1, (2.44)

64


where κ = 1 or 2 according to whether V is symmetric in K ,[·, ·] or not, then the spectrum of A 1 is likewise separated into two parts σ1 (A 1 ) and σ2 (A 1 ), Γ ⊂ ρ (A 1 ) and the results of Theorem 1.35 hold.

Proof. With Theorem 2.48 the proof is analogous to the proof of Corol■ lary 2.42. Now we are able to formulate the analogue of Theorem 2.43 for pseudoFriedrichs extensions: Theorem 2.51. Let A 0 be a self-adjoint and V a symmetric operator in a Krein space K ,[·, ·] such that D (V ) ⊂ D (A 0 ) and |[V x, x]| ≤ α x 2 + β [J | A 0 | x, x],

x ∈ D (V ),

where α ≥ 0 and 0 ≤ β < 1/2. Suppose that D (V ) is a core of | A 0 |1/2 and A 0 is also self-adjoint in the Hilbert space K , (·, ·) . Then the claims of Theorem 2.43 hold. Proof. Using the preceding results, the proof is analogous to the proof of Theorem 2.43. ■ Remark 2.52. A similar version of Theorem 2.51 has also been proved in [Ves72b], see Theorem 3 therein (compare also [Ves72a]). In comparison to Theorem 2.51, Theorem 3 in [Ves72b] requires further assumptions on the constants α, β and the length of the spectral gaps but shows that in addition to the reality of σ(A 1 ) the pseudo-Friedrichs extension A 1 is similar to an operator which is self-adjoint in a Hilbert space, or, equivalently, A 1 is of scalar type, see [Ves72a], [Ves72b] and [DS88, Chapter XV].

Chapter 3 Examples In this chapter we present examples and applications for the main results of Chapter 1 on relatively bounded perturbations of self-adjoint operators in Krein spaces. While the family of operators A ε , ε ∈ [0,1], of the first example (Section 3.1) is also self-adjoint in a Hilbert space, which includes the spectrum of A ε being real, the operators A ε of the remaining two examples (Section 3.2 and Section 3.3) are not. The family of operators considered in Section 3.3 was introduced by E. Caliceti and S. Graffi in [CG05].

3.1 Example 1 In the Hilbert space L2 (−L, L), where L ∈ R+ , besides the standard positive definite inner product f (x)g(x) dx, f , g ∈ L2 (−L, L), ( f , g) = [−L,L]

we consider the indefinite inner product defined by (3.1) [ f , g] := f (x)g(− x) dx, f , g ∈ L2 (−L, L).

2

[−L,L]

Then L (−L, L), [·, ·] is a Krein space, a fundamental decomposition is given by ˙ ] L2o (−L, L), L2e (−L, L) [+

(3.2)

where L2e (−L, L) and L2o (−L, L) are the sets of even and odd functions of L2 (−L, L), respectively. We may give an explicit formula for the fundamental symmetry J in the Krein space L2 (−L, L), [·, ·] corresponding to decomposition (3.2). Each f ∈ L2 (−L, L) can be decomposed as a sum f = f e + f o with f e (x) :=

1 f (x) + f (− x) , 2

f o (x) :=

1 f (x) − f (− x) , 2

J. Nesemann, PT-Symmetric Schrödinger Operators with Unbounded Potentials, DOI 10.1007/978-3-8348-8327-8_4, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

66

Chapter 3 Examples

where f e ∈ L2e (−L, L) and f o ∈ L2o (−L, L). For each such f we have J f (x) = f e (x) − f o (x) = f e (− x) + f o (− x) = f (− x).

(3.3)

Definition 3.1. In the Krein space L2 (−L, L), [·, ·] , where L ∈ R+ , we define the operators A 0 and V by D (A 0 ) := f ∈ L2 (−L, L) : f ∈ AC 2 (−L, L), f ∈ L2 (−L, L), f (−L) = f (L) = 0 ,

A 0 f := −

d2 f, dx2

f ∈ D (A 0 ),

and

D (V ) := f ∈ L2 (−L, L) : f ∈ AC(−L, L), f ∈ L2 (−L, L), f (−L) = f (L) = 0 ,

V f := i

d f, dx

f ∈ D (V ).

We define the family of operators A ε , ε ∈ [0,1], by A ε := A 0 + εV , ε ∈ [0,1]. Proposition 3.2. The operator A 0 from opera Definition 3.1 is a self-adjoint tor in the Hilbert space L2 (−L, L),(·, ·) and in the Krein space L2 (−L, L), [·, ·] . 2 Proof. It is well-known that A 0 is self-adjoint in the Hilbert space L (−L, L), (·, ·) , see [Kat95, Example V.3.25]. Hence, by equation (1.6) and by the definition of A 0 , we have A [0∗] f (x) = J A 0 J f (x) = J A 0 f (− x) = A 0 f (x) for f ∈ D (A 0 ). [∗] By (3.3), 2D (A 0 ) = D (JA 0 J) = D A 0 , that is, A 0 is a self-adjoint in the Krein ■ space L (−L, L), [·, ·] .

Proposition V from Definition 3.1 is symmetric in the 3.3. The operator Krein space L2 (−L, L), [·, ·] . 2 Proof. It is well-known that V is symmetric in the Hilbert space L (−L, L), (·, ·) , see [Kat95, Example V.3.14]. Thus, by equation (1.6),

V [∗] f (x) = JV ∗ J f (x) = JV J f (x) = JV f (− x) = V f (x), f ∈ D (V ), and hence V is symmetric in the Krein space L2 (−L, L), [·, ·] .

■

Remark 3.4. It is a simple exercise to show that the spectrum of the unperturbed operator A 0 consists entirely of the eigenvalues: 2 nπ σ(A 0 ) = λ0n : n ∈ N , λ0n := , n ∈ N. 2L

3.1 Example 1

67

The distance of two consecutive eigenvalues is growing linearly with n: n2 π2 (n − 1)2 π2 (2n − 1)π2 − = , n ∈ N, n ≥ 2. 4L2 4L2 4L2 Hence for δn = dist λ0n , σ(A 0 )\{λ0n } , n ∈ N, we establish δ1 = λ02 − λ01 and δn = λ0n − λ0n−1 , n ∈ N, n ≥ 2. λ0n − λ0n−1 =

Proposition 3.5. For [a, b] ⊂ R and n ∈ N define D (D) := f ∈ L2 (a, b) : f ∈ AC(a, b), f ∈ L2 (a, b), f (a) = f (b) ,

D f := −i

d f, dx

f ∈ D (D).

Then, for every n ∈ N, D n is relatively bounded with respect to D 2n such that for any τ > 0 we have (3.4)

n 2

D f ≤ 1 f 2 + τ D 2n f 2 , 4τ

f ∈ D D 2n .

Proof. Let n ∈ N and let f ∈ D D 2n . By the Cauchy-Schwarz inequality and n 2 since D is self-adjoint in the Hilbert space L (a, b), (·, ·) , we have

n 2

D f = (D n f , D n f ) = (D 2n f , f ) ≤ D 2n f f ≤ 1 f 2 + τ D 2n f 2 , 2τ 2

For τ := τ/2 we obtain the desired result.

τ > 0.

■

Remark 3.6. Proposition 3.5 is special case of a more general result; it is possible to show by induction that for every m, n ∈ N, m < n, D m is relatively compact with respect to D n . Here we restrict us to the case n = 2m because in the following we are particularly interested in the constants 1/(4τ) and τ of (3.4). Remark 3.7. Since A 0 and V are self-adjoint in the Hilbert space L2 (−L, L), (·, ·) , see Propositions 3.2 and 3.3, and since V is A 0 -compact by Proposi tion 3.5, A 1 = A 0 + V is self-adjoint in the Hilbert space L2 (−L, L),(·, ·) . Consequently, the spectrum of A 0 is real. Nevertheless, we show that also the assumptions of Theorem 1.44 (ii) are satisfied. By Proposition 3.2, A 0 is self-adjoint in the Krein space L2 (−L, L),[·, ·]. According to Propositions 3.2, 3.3 and 3.5, A ε is self-adjoint in L2 (−L, L),[·, ·] .

68

Chapter 3 Examples

We have to check if condition (1.24) is satisfied. Set κ := π/(2L). By Remark 3.4, to Proposiwe have δ1 = 3κ2 and δ n = (2n − 1)κ2 , n ∈ N, n ≥ 2. According 2 + 2n − 1)/2 and tion 3.5, we can choose α = κ and β = 1/(4 κ ), and α = ( κ n 1 1 n βn = 1/(2κ n2 + 2n − 1), n ∈ N, n ≥ 2. Hence 1 αn + βn δn + |λ0n | n∈N δ n 1 κ n2 + 2n − 1 1 2 2 = sup + (2n − 1)κ + (nκ) 2 2 n∈N, n≥2 (2n − 1)κ 2κ n2 + 2n − 1 n2 + 2n − 1 1 n2 + + = sup 2κ(2n − 1) n∈N, n≥2 2κ n2 + 2n − 1 2κ(2n − 1) n2 + 2n − 1 n2 + 2n − 1 = sup n∈N, n≥2 κ(2n − 1) 7 = < ∞, 3κ

γ : = sup

where we have used the fact that the function

n2 + 2n − 1 κ(2n − 1)

is monotonously decreasing for n ≥ 2. If we choose ε0 = 1/(2γ), i.e., 3κ 3π ε0 = = , 2 7 4 7L then (1.24) holds for all ε ∈ [0, ε0 ) ∩ [0,1], where ε0 > 1 is possible. In fact, the eigenvalues of A 1 are obtained by solving the differential equation (3.5)

d2 d f −i f +λf = 0 dx dx2

on the interval [−L, L] with the boundary conditions given by Definition 3.1. The general solution of (3.5) is (3.6)

i

f (x) := γ1 e 2

1− 4λ+1 x

i

+γ2 e 2

1+ 4λ+1 x

,

3.2 Example 2

69

for x ∈ [−L, L] and λ ∈ C, with arbitrary constants γ i , i = 1, 2. Substituting the boundary conditions f (−L) = f (L) = 0 into (3.6), we obtain the system of equations i i e− 2 L 1− 4λ+1 e− 2 L 1+ 4λ+1 γ1 0 = i i L 1 − 4 λ + 1 L 1 + 4 λ + 1 γ 0 2 e2 e2 which leads to non-trivial solutions if the determinant equals 0: 2 nπ 1 − , 2i sin L 4λ + 1 = 0 ⇐⇒ λ = 2L 4

n ∈ N0 .

If λ = −1/4, then f is the trivial solution, and thus the eigenvalues of A 1 are λ0n =

nπ 2L

2

1 − , 4

n ∈ N.

3.2 Example 2 Definition 3.8. In the Krein space L2 (−L, L), [·, ·] , where L ∈ R+ and [·, ·] as in (3.1), we define the operators A 0 and V by D (A 0 ) := f ∈ L2 (−L, L) : f ∈ AC 4 (−L, L), f (4) ∈ L2 (−L, L), f (−L) = f (L) = f (−L) = f (L) = 0 ,

A 0 f :=

d4 f, dx4

f ∈ D (A 0 ),

and D (V ) := f ∈ L2 (−L, L) : f ∈ AC 2 (−L, L), f ∈ L2 (−L, L), f (−L) = f (L) = 0 , d d V f := i h(x) f , f ∈ D (V ), dx dx

where h ∈ C 2 (−L, L) is a real-valued function having the following properties: h(− x) = − h(x)

for x ∈ [−L, L],

h(x) > 0

for x ∈ (0, L).

We define the family of operators A ε := A 0 + εV , ε ∈ [0,1]. operaProposition 3.9. The operator A 0 from Definition 3.8 is a self-adjoint tor in the Hilbert space L2 (−L, L), (·, ·) and in the Krein space L2 (−L, L), [·, ·] .

70

Chapter 3 Examples

Proof. The calculations are similar to those of the preceding example, com■ pare Proposition 3.2. Proposition 3.10. The operator V from Definition 3.8 is symmetric in the Krein space L2 (−L, L), [·, ·] . Proof. Integration by parts yields for f , g ∈ D (V ): d d [V f , g] = i h(x) f (x) g(− x) dx dx [−L,L] dx d d = −i h(x) f (x) g(− x) dx dx dx [−L,L] d d f (x)h(− x) g(− x) dx =i dx [−L,L] dx d d = f (x) i h(− x) g(− x) dx dx dx [−L,L]

= [f , V g].

■

Proposition 3.11. Let A 0 be the operator defined in Definition 3.8. Then the spectrum of A 0 consists of the eigenvalues λ0n :=

nπ 2L

4

,

n ∈ N.

Proof. Since A 0 is the square of A 0 from Definition 3.1 of Example 1, the statement follows from the spectral mapping theorem, see, e.g., [GGK90, Theorem 3.3]. ■ Remark 3.12. For the distance of two consecutive eigenvalues of A 0 , we have: 3 4n − 6n2 + 4n − 1 π4 n4 π4 (n − 1)4 π4 0 0 − = , n ∈ N, n ≥ 2. (3.7) λn − λn−1 = (2L)4 (2L)4 (2L)4 Remark 3.13. Let A 0 and V be linear operators as in Definition 3.8. Then V is relatively bounded with respect to A 0 with relative bound 0 such that for every τ > 0 inequality (1.7) holds with ατ = c/τ and βτ = τ, where c > 0 is constant.

3.2 Example 2

71

Proof. Let f ∈ D (A 0 ) ⊂ D (V ). Since V f = ih (x)

d d2 f + ih(x) 2 f , dx dx

we have

2

d

+ sup |h| d f f V f ≤ sup |h |

dx

dx2 and hence, by Proposition 3.5, we obtain the desired result.

■

Theorem 3.14. Consider the family of operators A ε , ε ∈ [0,1], from Defini tion 3.1 in the Krein space L2 (−L, L),[·, ·] . Then there exists an ε0 ∈ (0,1] such that for all ε ∈ [0, ε0 ) the spectrum of A ε = A 0 + εV consists of simple and real eigenvalues and which are of definite type. A is self-adjoint in the Hilbert space L2 (−L, L), Proof. By Proposition 3.9, 20 (·, ·) and in Krein space L (−L, L),[ to Proposition 3.10 and ·, ·] . According Remark 3.13, V is symmetric in L2 (−L, L),[·, ·] and A 0 -bounded with A 0 bound 0. To complete the proof we have to check if condition (1.24) is satisfied. The distance δn := dist λ0n , σ(A 0 )\{λ0n } equals the distance of two consecutive eigenvalues. Set κ := π/(2L). By equation (3.7), we have δ1 = 15κ4 3 and δn = 4n − 6n2 + 4n − 1 κ4 , n ∈ N, n ≥ 2. By Remark 3.13, we can choose α1 = 4κ2 and β1 = c/(4κ2 ), and

c αn = κ2 c n4 + 4n3 − 6n2 + 4n − 1 and βn = κ2 c n4 + 4n3 − 6n2 + 4n − 1

for n ∈ N, n ≥ 2. Hence 1 0 γ : = sup α n + β n δ n + |λ n | n∈N δ n ⎧ ⎫

⎪ ⎨ 4(c + 1) ⎬ 2 c n4 + 4n3 − 6n2 + 4n − 1 ⎪ = max , sup ⎪ ⎪ κ2 (4n3 − 6n2 + 4n − 1) ⎩ 15κ2 n∈N n≥2 ⎭ " # 4(c + 1) 2 31c = max < ∞, , 15κ2 15κ2 where we have used the fact that the function

2 c n4 + 4n3 − 6n2 + 4n − 1 κ2 4n3 − 6n2 + 4n − 1

72

Chapter 3 Examples

is monotonously decreasing for n ≥ 2. If we choose ε0 = 1/(2γ), i.e., ε0 = min

15κ2 15κ2 15π2 15π2 , , , = min 8(c + 1) 4 31c 32L2 (c + 1) 16L2 31c

then (1.24) holds for all ε ∈ [0, ε0 ) ∩ [0, 1], where ε0 > 1 is possible. Now the ■ claim follows from Theorem 1.44 (ii).

3.3 A Class of Schrödinger Operators with Relatively Bounded Complex Potentials and Real Spectrum In this paragraph we consider operators induced by the differential expression d2 A ε = − 2 + VP + ε VQ , ε ∈ [0, 1], dx in the Krein space L2 (R), [·, ·] . Here VP and VQ are multiplication operators with P(x) and iQ(x), respectively, where P(x) is a real, even polynomial of degree 2p, p ≥ 1, with lim|x|→∞ P(x) = ∞, and Q(x) is a real, odd polynomial of degree 2q − 1, q ≥ 1, such that p > 2q. This class of operators was considered in [CG05], where the results were proved by means of perturbation theory for linear operators. Here we are able to show, it is also an example for the results of Chapter 1. Definition 3.15. Consider the Krein space L2 (R),[·, ·] , with [·, ·] given by (3.8) [f , g] := f (x)g(− x) dx, f , g ∈ L2 (R). R

Let P : R → R be an even polynomial of degree 2p, p ≥ 1, such that lim P(x) = ∞

| x|→∞

and let Q : R → R be an odd polynomial of degree 2q − 1, q ≥ 1, such that p > 2q. Then we define the multiplication operators VP and VQ by D (VP ) := f ∈ L2 (R) : P f ∈ L2 (R) ,

VP f (x) := P(x) f (x),

f ∈ D (VP ),

Operators with Relatively Bounded Complex Potentials and Real Spectrum

and

73

D (VQ ) := f ∈ L2 (R) : iQ f ∈ L2 (R) ,

VQ f (x) := iQ(x) f (x),

f ∈ D (VQ ).

Furthermore, we define the family of operators A ε , ε ∈ [0, 1], D (A ε ) := f ∈ L2 (R) : f ∈ AC 2 (R), − f + VP f + εVQ f ∈ L2 (R) , d2 A ε f := − 2 + VP + εVQ f , f ∈ D (A ε ), ε ∈ [0, 1]. dx

Proposition 3.16. The operator A 0 defined in Definition 3.15 is self-adjoint in the Hilbert space L2 (R), (·, ·) and self-adjoint in the Krein space L2 (R),[·, ·] . (Here (·, ·) denotes the standard positive definite Hilbert space inner product on L2 (R)). Proof. By [BS91, Theorem 2.1.1], the minimal operator 0 A 0 := A 0 |C0∞ (R) is essentially self-adjoint in L2 (R),(·, ·) . Since 0 A 0 ⊂ 0 A 0 ⊂ A 0 , we have ∗ A ∗0 ⊂ 0 A 0 = 0 A 0 ⊂ A 0 and thus A 0 is self-adjoint in L2 (R), (·, ·) . Using the fact that P(x) is an even polynomial, integration by parts yields for f , g ∈ D (A 0 ):

d2 + V f (x)g(− x) dx P dx2 R d2 = − 2 f (x)g(− x) dx + P(x) f (x)g(− x) dx R dx R d2 = − f (x) 2 g(− x) dx + f (x)P(− x)g(− x) dx dx R R 2 d = f (x) − 2 + VP g(− x) dx dx R

[A 0 f , g] =

−

= [ f , A 0 g]. Hence A 0 is also symmetric in the Krein space L2 (R), [·, ·] and, since D (A 0 ) = [∗] 2 D (J A 0 J) = D A 0 , A 0 is even self-adjoint in L (R),[·, ·] . ■

The following remark and its proof can be found in [BS91, Theorem 2.3.1] and [CG05, (1.3)]. Remark 3.17. The family of operators A ε , ε ∈ [0, 1], defined in Definition 3.15 has discrete spectrum. Moreover, the eigenvalues λ0n , n ∈ N, of the unper-

74

Chapter 3 Examples

turbed operator A 0 are simple, form an increasing sequence and satisfy the estimate p −1 2p λ0n = c 1 n p+1 + O n p+1 , n → ∞, (3.9) with some constant c 1 > 0; here O denotes the Landau symbol. The following proposition can be found in [CG05, Lemma 2.1]. Proposition 3.18. The multiplication operator VQ given by Definition 3.15 is relatively bounded with respect to A 0 with A 0 -bound 0. Moreover we have the following. For all n ∈ N there exist constants c 2 , c 3 > 0 such that p −1

VQ f ≤ c 2 n p+1 f +

(3.10)

c3 A 0 f , n

f ∈ D (A 0 ).

Proof. According to [Sim70], there exist γ1 , γ2 > 0 such that

2

d

(3.11)

dx2 f + VP f ≤ γ1 f + γ2 A 0 f , f ∈ D (A 0 ). Therefore it is sufficient to show that VQ f ≤ α n f + βn VP f ,

(3.12)

f ∈ D (VP ),

p −1

where α n = γ3 n p+1 and βn = γ4 /n for n ∈ N, γ3 , γ4 > 0, since then, by (3.11), VQ f ≤ α n + βn γ1 f + βn γ2 A 0 f , f ∈ D (VP ), which implies (3.10). By Lemma 1.13, inequality (3.12) is equivalent to VQ f 2 ≤ α 2n f 2 + β2n VP f 2 ,

(3.13)

f ∈ D (VP ).

Inequality (3.13) is implied by the pointwise inequality 2n + β2n P(x)2 − Q(x)2 , 0≤α

(3.14)

x ∈ R,

since (3.14) leads to α 2n f 2 − VQ f 2 + β2n VP f 2 , 2n + β2n P(x)2 − Q(x)2 | f (x)|2 dx = α 0≤ R

for f ∈ L2 (R). P(x) and Q(x) can be minorized and majorized, respectively, by homogeneous polynomials of degree 2p and 2q − 1, respectively. Thus, up


75

to an additive constant, which can be absorbed in the constant γ1 in (3.11), we can restrict ourselves to verify (3.14) for homogeneous polynomials P(x) and Q(x) of degree 2p and 2q − 1, respectively. Let βn = γ4 /n, n ∈ N, for some γ4 > 0. Then (3.14) is satisfied if 0≤α 2n +

(3.15)

γ24

n2

x4p − x4q−2 ,

x ∈ R.

Since there are only even powers, we can restrict ourselves to x ≥ 0. Set s :=

1 . 2(p − q) + 1

We consider two cases. First, let x ≥ n s . Hence x ≥ n s ⇐⇒ x4q−2 ≤ n−2 x4p ⇐⇒ 0 ≤ n−2 x4p − x4q−2 ,

x ∈ R+ 0,

and thus (3.15) is satisfied with α n = 0, n ∈ N, and γ4 = 1. Now let x < n s . Then we have x4q−2 < n(4q−2)s and thus (3.15) is satisfied with 2q−1

α n = n 2(p−q)+1

and

γ4 = 0.

By Definition 3.15, p > 2q which implies 2q − 1 p−1 < . 2(p − q) + 1 p + 1

■

Proposition 3.19. The multiplication operator VQ defined in Definition 3.15 is symmetric in the Krein space L2 (R), [·, ·] . Proof. Since Q(x) is a real, odd polynomial, we have iQ(− x) = −iQ(− x) = iQ(x),

x ∈ R.

Hence V is symmetric in L2 (R), [·, ·] :

[VQ f , g] = =

R R

for all f , g ∈ D (VQ ).

VQ f (x)g(− x)dx =

iQ(x) f (x)g(− x)dx f (x)iQ(− x)g(− x)dx = f (x)VQ g(− x)dx = [ f , VQ g], R

R

■

76

Chapter 3 Examples

Theorem 3.20. The family of operators A ε , ε ∈[0, 1], defined in Definition 3.15 is self-adjoint in the Krein space L2 (R), [·, ·] for all ε ∈ [0,1], and there exists an ε0 ∈ [0, 1] such that for all ε ∈ [0, ε0 ) the spectrum of A ε = A 0 + εV consists of simple real eigenvalues of definite type. and Proof. By Propositions 3.19 and 3.18, V is symmetric A 0 -bounded with A 0 -bound 0. Hence, since A 0 is self-adjoint in L2 (R), [·, ·] by Proposition 3.16, A ε = A 0 + εV is self-adjoint for all ε ∈ [0, 1] by Theorem 1.24. To complete the proof we have to check if condition (1.24) is satisfied. The distance δn := dist λ0n , σ(A 0 )\{λ0n } is the minimum of the distances between the eigenvalues λ0n and λ0n−1 , or λ0n and λ0n+1 , i.e., δ n = min λ0n − λ0n−1 , λ0n+1 − λ0n ,

n ∈ N, n ≥ 2.

Since 2p

p −1

2p

p −1

n p+1 − (n − 1) p+1 = n · n p+1 − (n − 1)(n − 1) p+1 ⎛ ⎞ pp−+11 p −1 n − 1 ⎠ = n p+1 ⎝ n − (n − 1) n ⎛ ⎞ pp−+11 pp−+11 p −1 n − 1 n − 1 ⎠ = n p +1 ⎝ n − n + n n ⎛ ⎞ 1 p−1 pp−+11 p +1 p −1 1 − n− n − 1 n ⎠, = n p +1 ⎝ + n n− 1 we have 2p p−1 2p p −1 λ0n − λ0n−1 = c 1 n p+1 − (n − 1) p+1 + O n p+1 − O (n − 1) p+1 ⎛ ⎞ 1 p −1 pp−+11 p +1 p −1 p −1 1 − n− n − 1 n ⎠ + O n p +1 , + = c 1 n p +1 ⎝ n n−1

n → ∞,

and thus λ0n − λ0n−1 p−1

n p+1

⎛ = c1 ⎝

1−

n−1 p−1 n n−1

p +1

n−1 + n

pp−+11

⎞

⎠+O 1 ,

n → ∞.


77

By l’Hospital’s rule,

lim

1−

n−→∞

n−1 p−1 n

p +1

n−1

= lim

n−→∞

= lim

n−→∞

=

p−1 n−1 p+1 −1 1 n n2 − 2 −n

− p+1

p−1

−2 p − 1 n − 1 p +1 p+1 n

p−1 , p+1

and we obtain lim sup n−→∞

λ0n − λ0n−1 p −1

p −1

< ∞,

n p +1

lim sup n−→∞

n p +1 λ0n − λ0n−1

< ∞.

Consequently, p −1

δn ∼ c 4 n p+1 ,

λ0n δn

∼ c 5 n,

n → ∞,

with constants c 4 , c 5 > 0, where ∼ stands for asymptotic equivalence (two ∞ sequences (s n )∞ n=1 and (t n ) n=1 are called asymptotically equivalent, in symp −1

bols, s n ∼ t n , if and only if s n ∈ O (t n ) and t n ∈ O (s n )). If we set αn := c 2 n p+1 and βn := c 3 /n, n ∈ N, with c 2 , c3 > 0 according to Proposition 3.18,then, by Proposition 3.18, 1 sup αn + βn δn + |λ0n | < ∞. n∈N δn Thus there exists an ε0 ∈ [0,1] such that (1.24) holds for all ε ∈ [0, ε0 ), and the ■ claim follows from Theorem 1.44 (ii). Remark 3.21. The last theorem has also been proved in works by E. Caliceti and S. Graffi (see [CG05, Theorem 1.1], compare also [Cal04], [CGS05], [Cal05] and [CCG06]). Remark 3.22. Similar operators with different assumptions on the polynomials P and Q are considered in works by Dorey et al. (see [DDT01b]) and Shin (see [Shi02]).

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