Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

Mantile, Andrea; Posilicano, Andrea; Sini, Mourad

doi:10.1016/j.jde.2015.11.026

Cited by 50 publications

(86 citation statements)

References 47 publications

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“…For proofs and more details on such realizations, we refer to [1,. In particular, by the results given there, hypothesis (49) or (50) hold for the models considered here, namely: (49) is satisfied in the case of "global" boundary conditions (i.e.…”

Section: Dirichlet and Neumann Boundary Conditions On σ ⊆ γmentioning

confidence: 77%

“…For generic elliptic selfadjoint operators with smooth coefficients, this strategy has been implemented in [1] to which we refer for the detailed proofs. The auxiliary operators G z are next defined by the duality:…”

Section: Preliminariesmentioning

confidence: 99%

“…In a recent paper [1], the complete family of self-adjoint elliptic operators with interface conditions assigned on a hypersurface in R n was realized. Derived from the abstract theory of selfadjoint extensions of restrictions developed in [2][3][4][5], our approach leads to Kreȋn type formulae for the resolvent difference between the perturbed operator and the corresponding free selfadjoint model with domain H 2 (R n ).…”

Section: Introductionmentioning

confidence: 99%

“…This is a relevant point for the interface perspective of studying the scattering problem. Moreover, while some sub-families of extensions (mainly those concerned with the δ or δ interface conditions) have been largely investigated by using quadratic form or quasi-boundary triple techniques (see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]), for others models presented in [1], and in particular those concerned with local interface conditions of Dirichlet and Neumann type, a rigorous analysis was not previously given.…”

Section: Introductionmentioning

confidence: 99%

“…In this framework, we recall the basic results needed to construct the whole family of singular perturbations and then focus on the explicit examples of "global" and "local" Dirichlet-and Neumann-type boundary conditions. For the detailed proofs, we refer to [1].…”

Section: Introductionmentioning

confidence: 99%

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Laplacians with singular perturbations supported on hypersurfaces

Mantile¹,

Posilicano²

2016

Nanosystems: Phys. Chem. Math.

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We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Kreȋn-like resolvent formulae where the reference operator coincides with the free selfadjoint Laplacian in R n , providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.

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Section: Dirichlet and Neumann Boundary Conditions On σ ⊆ γmentioning

confidence: 77%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Laplacians with singular perturbations supported on hypersurfaces

Mantile¹,

Posilicano²

2016

Nanosystems: Phys. Chem. Math.

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Construction of self‐adjoint differential operators with prescribed spectral properties

Behrndt

Khrabustovskyi

2022

Mathematische Nachrichten

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In this expository article some spectral properties of self‐adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

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Boundary Value Problems, Weyl Functions, and Differential Operators

Behrndt

Hassi

Snoo

2020

Monographs in Mathematics

146

227

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The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

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Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

Cited by 50 publications

References 47 publications

Laplacians with singular perturbations supported on hypersurfaces

Laplacians with singular perturbations supported on hypersurfaces

Construction of self‐adjoint differential operators with prescribed spectral properties

Boundary Value Problems, Weyl Functions, and Differential Operators

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