On the Number of Negative Eigenvalues of a Schrödinger Operator with Point Interactions

Ogurisu, Osamu

doi:10.1007/s11005-008-0258-3

Cited by 16 publications

(22 citation statements)

References 4 publications

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“…Finally in this section, we emphasize that L is a discrete analogue of the point-interaction Hamiltonian H on R d , which is discussed in [1,2,9,10] and in a lot of literature; H is a Schrödinger operator with pseudo-potentials…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

Hayashi

Higuchi

Nomura³

et al. 2016

Lett Math Phys

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On the d-dimensional lattice Z d and the r-regular tree T r , an exact expression for the number of discrete eigenvalues of a discrete Laplacian with a finitely supported potential is described in terms of the support and the intensities of the potential on each case. In particular, the number of eigenvalues less than the infimum of the essential spectrum is bounded by the number of negative intensities.Mathematics Subject Classifications. 39A12, 47A75, 34L40.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We can prove Theorem 4.3 by virtue of Theorem 4.1, Theorem 1.1 and the following theorem about H; Theorem 4.5 (Theorem 1 in [10]). We have that…”

Section: The Number Of the Discrete Eigenvaluesmentioning

confidence: 99%

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

Hayashi

Higuchi

Nomura³

et al. 2016

Lett Math Phys

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“…In the case of a Schrödinger operator with δ-interactions, the problem of estimating the number of negative eigenvalues is rather nontrivial, even in the case of finitely many point interactions. For further details we refer to, e.g., [3], [4], [22], [31], [33], and [38].…”

Section: Negative Spectrummentioning

confidence: 99%

One-dimensional Schrödinger operators withδ′-interactions on Cantor-type sets

Eckhardt

Kostenko

Malamud

2014

Journal of Differential Equations

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Abstract. We introduce a novel approach for defining a δ ′ -interaction on a subset of the real line of Lebesgue measure zero which is based on SturmLiouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ ′ -interactions concentrated on sets of complicated structures.

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“…Schrödinger operators with singular δ-type interactions supported on discrete sets, curves and surfaces are used for the description of quantum mechanical systems with a certain degree of idealization. The spectral properties of Schrödinger operators with δ and δ ′ -interactions were investigated in numerous mathematical and physical articles in the recent past; we mention only [BN11, KM10,MS12,O10] for interactions on point sets, [CK11, EI01, EK08, EN03, EP12, K12, KV07] on curves, and [AKMN13, BLL13, EF09, EK03] for interactions on surfaces. For a survey and further references we refer the reader to [E08] and to the standard monograph [AGHH].…”

Section: Introductionmentioning

confidence: 99%

Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

2014

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Abstract. We investigate Schrödinger operators with δ and δ ′ -interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrödinger operators with δ and δ ′ -interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ ′ -interactions.

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On the Number of Negative Eigenvalues of a Schrödinger Operator with Point Interactions

Cited by 16 publications

References 4 publications

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

One-dimensional Schrödinger operators withδ′-interactions on Cantor-type sets

Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

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