The Generalized Birman-Schwinger Principle

Behrndt, Jussi; Elst, A. F. M. ter; Gesztesy, Fritz

doi:10.48550/arxiv.2005.01195

Cited by 5 publications

(5 citation statements)

References 70 publications

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“…Let us mention Kato's pioneering work [37] and the work by Konno and Koroda [39]. As some of the more recent articles on the topic we mention works by Gesztesy et al [30], Latushkin and Sukhtayev [42], Frank [28] and of Behrndt, ter Elst and Gesztesy [3]. The assumptions on A, B and H 0 made in these works are not uniform but vary from paper to paper.…”

Section: Assumptions and Notationsmentioning

confidence: 99%

The abstract Birman-Schwinger principle and spectral stability

Hansmann¹,

Krejčiřı́k²

2020

Preprint

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We discuss abstract Birman-Schwinger principles to study spectra of self-adjoint operators subject to small non-self-adjoint perturbations in a factorised form. In particular, we extend and in part improve a classical result by Kato which ensures spectral stability. As an application, we revisit known results for Schrödinger and Dirac operators in Euclidean spaces and establish new results for Schrödinger operators in three-dimensional hyperbolic space.

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Section: Assumptions and Notationsmentioning

confidence: 99%

The abstract Birman-Schwinger principle and spectral stability

Hansmann¹,

Krejčiřı́k²

2020

Preprint

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“…Remark 3.6. The operator Q of (33) often arises in spectral analysis of perturbations of the form (28) (see [16,18,24]); it is an operator-valued Herglotz function [12] which is well known for its role in the Birman-Schwinger principle and the spectral shift, see [1,2,13,20] and references therein. It is the Birman-Schwinger principle (see, e.g.…”

Section: The Main Resultsmentioning

confidence: 99%

“…Assume that below its essential spectrum, S has an eigenvalue ı with the eigenfunction f . Consider further a self-adjoint non-negative 1 perturbation operator P such that Pf D 0. This is a perturbation "along" the eigenfunction f : f is also an eigenfunction of the perturbed operator H WD S C P with eigenvalue ı .…”

Section: Introductionmentioning

confidence: 99%

Spectral shift via “lateral” perturbation

Berkolaiko

Kuchment

2022

J. Spectr. Theory

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We consider a compact perturbation H 0 D S C K 0 K 0 of a self-adjoint operator S with an eigenvalue ı below its essential spectrum and the corresponding eigenfunction f . The perturbation is assumed to be "along" the eigenfunction f , namely K 0 f D 0. The eigenvalue ı belongs to the spectra of both H 0 and S . Let S have more eigenvalues below ı than H 0 ; is known as the spectral shift at ı . We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue ı in the spectrum of H.K/ D S C K K. We show that the eigenvalue as a function of K has a critical point at K D K 0 and the Morse index of this critical point is the spectral shift . A version of this theorem also holds for some non-positive perturbations.Dedicated to the memory of Misha Shubin, a wonderful mathematician and person 2020 Mathematics Subject Classification. 47A10, 47A55, 47B25.

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“…Item (ii) can be seen as a 'boundary value' version of the Birmann-Schwinger principle (see e.g. [6,32] and references therein). The proof of the next proposition is a quite straightforward extension to that of [23,Lemma 4.1], where the result is proven for dissipative operators.…”

Section: Spectral Singularitiesmentioning

confidence: 99%

Spectral decomposition of some non-self-adjoint operators

Faupin¹,

Frantz²

2022

Preprint

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We consider non-self-adjoint operators in Hilbert spaces of the form H = H0 + CW C, where H0 is self-adjoint, W is bounded and C is a metric operator, C bounded and relatively compact with respect to H0. We suppose that C(H0 − z) −1 C is uniformly bounded in z ∈ C \ R. We define the spectral singularities of H as the points of the essential spectrum λ ∈ σess(H) such that C(H ± iε) −1 CW does not have a limit as ε → 0 + . We prove that the spectral singularities of H are in one-to-one correspondence with the eigenvalues, associated to resonant states, of an extension of H to a larger Hilbert space. Next, we show that the asymptotically disappearing states for H, i.e. the set of vectors ϕ such that e ±itH ϕ → 0 as t → ∞, coincide with the generalized eigenstates of H corresponding to eigenvalues λ ∈ C, ∓Im(λ) > 0. Finally, we define the absolutely continuous spectral subspace of H and show that it satisfies Hac(H) = Hp(H * ) ⊥ , where Hp(H * ) stands for the point spectrum of H * . We thus obtain a direct sum decomposition of the Hilbert spaces in terms of spectral subspaces of H. One of the main ingredients of our proofs is a spectral resolution formula for a bounded operator r(H) regularizing the identity at spectral singularities. Our results apply to Schrödinger operators with complex potentials.

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The Generalized Birman-Schwinger Principle

Cited by 5 publications

References 70 publications

The abstract Birman-Schwinger principle and spectral stability

The abstract Birman-Schwinger principle and spectral stability

Spectral shift via “lateral” perturbation

Spectral decomposition of some non-self-adjoint operators

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