Schrödinger-Operators with Singular Interactions

Brasche, Johannes F.; Exner, Pavel; Kuperin, Yu. A.; Šeba, P.

doi:10.1006/jmaa.1994.1188

Cited by 202 publications

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“…We also recall that the essential spectrum of the operator A δ,α is [0, ∞) and that the negative spectrum consists of finitely many eigenvalues of finite multiplicity; see [8,10]. In the following we fix some point η such that…”

Section: Schrödinger Operators With δ-Interactions On Hypersurfacesmentioning

confidence: 99%

“…where δ C denotes the δ-distribution supported on C. A precise definition of the self-adjoint operator A δ,α in terms of boundary or interface conditions is given at the beginning of Section 3 below; see also [8,10] for an equivalent definition via quadratic forms. Schrödinger operators with δ-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [1] and [23], to the review article [18] and to [2, 3, 8, 16, 17, 19-22, 24, 25, 32, 33, 36, 37, 42] for a small selection of related papers on spectral analysis of such operators.…”

Section: Introductionmentioning

confidence: 99%

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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: Schrödinger Operators With δ-Interactions On Hypersurfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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“…There is, of course, a considerable amount of previous work on these issues; see, for example, [1,3,5,6] and the references cited therein. Here, we will follow the approach of [3].…”

Section: Schrödinger Operators With Measuresmentioning

confidence: 99%

The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials

Remling¹

1998

Communications in Mathematical Physics

104

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Abstract. This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and DenisovRakhmanov type theorems.In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.

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“…One can fulfil the hypothesis that the operator 1/γ − R m,m (z) is invertible easily by choosing sufficiently large negative z; it follows from the next lemma, see [BEKŠ,corollary 2.2]. In the following, T p,q denotes the norm of an operator T acting from…”

Section: Krein-like Formulamentioning

confidence: 99%

Approximation by point potentials in a magnetic field

Ozanova¹

2006

J. Phys. A: Math. Gen.

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We discuss magnetic Schrödinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrödinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to calculate discrete spectra; we illustrate it by numerical calculations in the case when the potential is supported by a circle.

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Schrödinger-Operators with Singular Interactions

Cited by 202 publications

References 0 publications

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials

Approximation by point potentials in a magnetic field

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