Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains

Gesztesy, Fritz; Mitrea, Marius

doi:10.1090/pspum/079/2500491

Cited by 81 publications

(148 citation statements)

References 79 publications

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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: 68%

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: 68%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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“…An extension of the theory to relations can be found in the work of Malamud and Mogilevskii [21,22]. For related recent results, particularly in the context of PDEs, we refer to the works of Alpay and Behrndt [1], Behrndt and Langer [3], Brown, Grubb, Wood [6], Gesztesy and Mitrea [10,11,12] and also to Posilicano [24,25] and Post [26].…”

Section: 1)mentioning

confidence: 99%

The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

Brown

Hinchcliffe

Marletta

et al. 2009

Integr. equ. oper. theory

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Abstract. In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M -function see the same singularities as the resolvent of a certain restriction A B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S andS such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M -function is analytic. We present three examples -one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -which together indicate that the abstract results are probably best possible.

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“…[1][2][3][4][5][6][7][8][9][10][11][12]16,[18][19][20]23,24] for a small selection of papers of analytic nature. In the following let be a bounded Lipschitz domain in R n , where n ≥ 2, with boundary C and consider the differential expression L = − + V on with V ∈ L ∞ ( ) real valued.…”

Section: Introductionmentioning

confidence: 99%

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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Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains

Cited by 81 publications

References 79 publications

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

Dirichlet-to-Neumann maps on bounded Lipschitz domains

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