On a generic topological structure of the spectrum to one-dimensional Schrödinger operators with complex limit-periodic potentials

Leonid Zelenko

doi:10.1007/s00020-003-1239-7

On a generic topological structure of the spectrum to one-dimensional Schrödinger operators with complex limit-periodic potentials

Leonid Zelenko

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that for a massive set of limit-periodic complex-valued potentials V(x) (x ∈ IR) of Stepanov class, the spectra σ(H_v) of the corresponding one-dimensional Schrödinger operators H_v are 0-dimensional topological subspaces of the complex plane C. This is a generalization of a similar result obtained by J. Avron and B. Simon for the case of real-valued limit-periodic potentials. Our technique is based upon results of V. Tkachenko concerning an inverse spectral problem for the Hill operator with a complex-valued potential. We also make use of some facts from Differential Topology.

Original language	English
Pages (from-to)	393-430
Number of pages	38
Journal	Integral Equations and Operator Theory
Volume	50
Issue number	3
DOIs	https://doi.org/10.1007/s00020-003-1239-7
State	Published - Nov 2004

Bibliographical note

Funding Information:
Supported by the KAMEA Project for Scientific Absorption in Israel and partially by Grant no. 585/00 from the Israel Science Foundation.

Keywords

Generic spectrum
Limit-periodic potential
Massive set
Schrödinger operator

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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10.1007/s00020-003-1239-7

Cite this

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AB - We prove that for a massive set of limit-periodic complex-valued potentials V(x) (x ∈ IR) of Stepanov class, the spectra σ(Hv) of the corresponding one-dimensional Schrödinger operators Hv are 0-dimensional topological subspaces of the complex plane C. This is a generalization of a similar result obtained by J. Avron and B. Simon for the case of real-valued limit-periodic potentials. Our technique is based upon results of V. Tkachenko concerning an inverse spectral problem for the Hill operator with a complex-valued potential. We also make use of some facts from Differential Topology.

KW - Generic spectrum

KW - Limit-periodic potential

KW - Massive set

KW - Schrödinger operator

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JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

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ER -

On a generic topological structure of the spectrum to one-dimensional Schrödinger operators with complex limit-periodic potentials

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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