Laplacians on periodic graphs with guides

Saburova, Natalia

doi:10.1016/j.jmaa.2017.06.039

Cited by 12 publications

(4 citation statements)

References 42 publications

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“…Laplacians on periodic graphs with non-compact perturbations and the stability of their essential spectrum were considered in [9], [44]. Korotyaev-Saburova [30], [31] considered Schrödinger operators with periodic potentials on periodic discrete graphs with by so-called guides, which are periodic in some directions and finitely supported in others. They described some properties of so-called guided spectrum.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Eigenvalues of periodic difference operators on lattice octant

Korotyaev

2019

Preprint

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Consider a difference operator H with periodic coefficients on the octant of the lattice. We show that for any integer N and any bounded interval I, there exists an operator H having N eigenvalues, counted with multiplicity on this interval, and does not exist other spectra on the interval. Also right and to the left of it are spectra and the corresponding subspaces have an infinite dimension. Moreover, we prove similar results for other domains and any dimension. The proof is based on the inverse spectral theory for periodic Jacobi operators.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Eigenvalues of periodic difference operators on lattice octant

Korotyaev

2019

Preprint

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“…For a non-compact perturbation, [23] studies the stability of their essential spectrum. For Schrödinger operators on periodic graphs perturbed by guides, graphs which are periodic in some directions and finite in other ones, we refer to [17,18].…”

Section: Introductionmentioning

confidence: 99%

Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

Richard,

Tsuzu

2021

Preprint

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In this paper we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. A special attention is paid to perturbations obtained by the addition of an infinite number of edges, and / or by the removal of a finite number of them, but perturbations of the underlying measures and perturbations by the addition of a multiplication operator are also considered. The description of the nature of the spectrum of the resulting operators, and the existence and completeness of the wave operators are standard outcomes for these investigations.

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“…The spectrum of Laplacians on the lattice Z d with pendant edges was studied in [Su13]. In [KS17] the authors considered Laplace operators on periodic graphs perturbed by guides, i.e., graphs that are periodic in some directions and finite in other ones. A procedure of creating gaps in the spectrum of Laplacians on an infinite graph G by attaching a fixed finite graph G o to each vertex of G was presented in [AS00].…”

Section: Introductionmentioning

confidence: 99%

Spectrum of Schrödinger operators with potential waveguides on periodic graphs

Saburova

Post

2019

2019 Days on Diffraction (DD)

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We consider discrete Schrödinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. We perturb a periodic graph by adding edges in a periodic way (without changing the vertex set) and show that if the length of the added edges tends to infinity, then the perturbed graph is asymptotically isospectral to some periodic graph of a higher dimension but without long edges. We also obtain a criterion for the perturbed graph to be not only asymptotically isospectral but just isospectral to this higher dimensional periodic graph. One of the simplest examples of such asymptotically isospectral periodic graphs is the square lattice perturbed by long edges and the cubic lattice. We also get asymptotics of the endpoints of the spectral bands for the Schrödinger operator on the perturbed graph as the length of the added edges tends to infinity.

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Laplacians on periodic graphs with guides

Cited by 12 publications

References 42 publications

Eigenvalues of periodic difference operators on lattice octant

Eigenvalues of periodic difference operators on lattice octant

Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

Spectrum of Schrödinger operators with potential waveguides on periodic graphs

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