Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields

Verdìère, Yves Colin de; Torki-Hamza, Nabila; Truc, Francoise

doi:10.48550/arxiv.1011.6492

Cited by 3 publications

(4 citation statements)

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“…Some other criteria, based on commutators, are given in [GS] (see also [Gol]). Finally for the magnetic case, we mention the works [CTT2,Mil,Mil2,MiTr]. We point out that in the older work of [Aom] ones gives some characterization of possible self-adjoint extensions of a weighted discrete Laplacian in the limit point/circle spirit in the case of trees.…”

Section: 1mentioning

confidence: 98%

Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

Golenia

2014

Journal of Functional Analysis

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Section: 1mentioning

confidence: 98%

Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

Golenia

2014

Journal of Functional Analysis

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“…with the convention that A v,u = −A u,v , which makes H self-adjoint. For further details, the reader should consult [34,46,15,16].…”

Section: Magnetic Hamiltonian On Discrete Graphsmentioning

confidence: 99%

“…To give a sample, Harper [25] used the tight-binding model (discrete Laplacian) to describe the effect of the magnetic field on conduction (see also [30]). In mathematical literature, discrete magnetic Schrödinger operator was introduced by Lieb and Loss [34] and Sunada [45,46], and studied, among other sources, in [41,15,16] (see also [47] for a review).…”

Section: Introductionmentioning

confidence: 99%

Nodal count of graph eigenfunctions via magnetic perturbation

Berkolaiko

2013

Anal. PDE

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Abstract. We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a "zero"). It is known that the n-th eigenfunction has n − 1 + s such zeros, where the "nodal surplus" s is an integer between 0 and the number of cycles on the graph.We then perturb the Laplacian by a weak magnetic field and view the n-th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus s of the n-th eigenfunction of the unperturbed graph.

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“…Kato's inequality for ∆ σ as in (1.4), with w ≡ 1 and a ≡ 1, was proven in Dodziuk-Mathai [10] and used to study asymptotic properties of the spectrum of a certain discrete magnetic Schrödinger operator. For a study of essential self-adjointness of the magnetic Laplacian on a metrically non-complete graph, see Colin de Verdière, Torki-Hamza, and Truc [6]. A different model for discrete magnetic Laplacian was given by Sushch [28].…”

Section: Background Of the Problemmentioning

confidence: 99%