Schrödinger operators periodic in octants

Evgeny Korotyaev; Jacob Schach MØller

doi:10.1007/s11005-021-01402-4

Schrödinger operators periodic in octants

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Department of Mathematics

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Abstract

We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

Original language	English
Article number	55
Journal	Letters in Mathematical Physics
Volume	111
Issue	2
ISSN	0377-9017
DOIs	https://doi.org/10.1007/s11005-021-01402-4
Publication status	Published - Apr 2021

Keywords

Eigenvalues
Periodic Schrödinger operator
Spectral bands

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10.1007/s11005-021-01402-4

https://arxiv.org/pdf/1712.08893.pdf

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title = "Schr{\"o}dinger operators periodic in octants",

abstract = "We consider Schr{\"o}dinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schr{\"o}dinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schr{\"o}dinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.",

keywords = "Eigenvalues, Periodic Schr{\"o}dinger operator, Spectral bands",

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AU - Korotyaev, Evgeny

AU - MØller, Jacob Schach

PY - 2021/4

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N2 - We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

AB - We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

KW - Eigenvalues

KW - Periodic Schrödinger operator

KW - Spectral bands

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DO - 10.1007/s11005-021-01402-4

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JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

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M1 - 55

ER -

Schrödinger operators periodic in octants

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Keywords

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