Boundary pairs associated with quadratic forms

Post, Olaf

doi:10.1002/mana.201500048

Cited by 33 publications

(67 citation statements)

References 65 publications

(147 reference statements)

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“…Moreover, the spectral properties of A δ,α can be described with the help of the perturbation term (Θ δ,α − M (λ)) −1 . We mention that in the context of the more general notion of quasi boundary triples and their Weyl functions from [5,7] a similar approach as in this note and closely related results can be found in [8,9]; we also refer to [26,27,29,31,35,[38][39][40] for other methods in extension theory of elliptic differential operators.…”

Section: Introductionmentioning

confidence: 69%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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The self-adjoint Schrödinger operator A δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L 2 (R n ). The aim of this note is to construct a boundary triple for S * and a self-adjoint parameter Θ δ,α in the boundary space L 2 (C) such that A δ,α corresponds to the boundary condition induced by Θ δ,α . As a consequence the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of A δ,α in terms of the Weyl function and Θ δ,α .

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Section: Introductionmentioning

confidence: 69%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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“…For the case that λ ∈ C belongs to the resolvent set of both operators A D and A N such formulae are well known and can be found in e.g. [1,[7][8][9]12,18,19,23,30,31]. However, our aim is to show that the correspondence between (A D − λ) −1 and (A N − λ) −1 in terms of γ D (λ), γ N (λ), and the Dirichlet-to-Neumann map D(λ) and Neumann-to-Dirichlet map N (λ) is also valid if λ is an eigenvalue of one or both of the operators A D and A N .…”

Section: Dirichlet-to-neumann and Neumann-to-dirichlet Mapsmentioning

confidence: 96%

Dirichlet-to-Neumann maps on bounded Lipschitz domains

Behrndt

elst

2015

Journal of Differential Equations

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The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains.

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“…To set the stage, we also verify Krein-type resolvent formulas which relate the inverses (A − zI L 2 ( ) ) −1 and A * − zI L 2 ( ) −1 with the resolvents of the Dirichlet realizations A D and A D , respectively. For the self-adjoint case, such formulas are well-known and can be found, for example, in [1,8,9,11,15,29,30,48,58,59]. For dual pairs of elliptic differential operators we refer to [14], and for a more abstract operator theory framework, see [49] and [50].…”

Section: Hypothesis 37mentioning

confidence: 98%

“…In addition, we refer the reader to [1,2,[9][10][11][12][13][14][15][16]18,[20][21][22][23]37,38,[48][49][50][51][52][53][54][55][56][57][58][59][60], for more details, applications, and references on boundary triples and their Weyl-Titchmarsh functions.…”

Section: Appendix a Boundary Triples Weyl-titchmarsh Functions Andmentioning

confidence: 99%

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

Behrndt

Gesztesy

Holden

et al. 2016

Journal of Differential Equations

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We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh M-functions and Donoghuetype M-functions corresponding to closed extensions of symmetric operators belong to it.The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the ✩ JID:YJDEQ AID:8393 /FLA [m1+; v1.232; Prn:20/06/2016; 10:29] P.2 (1-37) 2 J. Behrndt et al. / J. Differential Equations ••• (••••) •••-••• corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh M-functions, respectively.

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Boundary pairs associated with quadratic forms

Cited by 33 publications

References 65 publications

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Dirichlet-to-Neumann maps on bounded Lipschitz domains

Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions

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