Construction of the essential spectrum for a multidimensional non-self-adjoint Schrödinger operator via the spectra of operators with periodic potentials, I

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Abstract

We describe the essential spectrum σe(H) of a multidimensional Schrödinger operator H with a complex-valued potential V(x) in terms of a family of Schrödinger operators {Hy}y∈ℝm with periodic potentials Vy(x) which approximate the potential V(x) at infinity in a sense. Under some conditions we prove that the set σe(H) coincides with the set Γ{Vy} of such points λ ∈ ℂ for which the family of norms {||Rλ(Hy)||}y∈ℝ is unbounded at infinity. Sometimes the set Γ{Vy} coincides with the set Σ{Vy} of limit points of the spectra σ(Hy) of the operators Hy for |y| → ∞. In this case we call the family {Vy(x)}x∈ℝm spectrally non-degenerate. We find some conditions of the spectral non-degeneracy. To this end we carry out an estimation of resolvents of the operators Hy with the help of generalized perturbation determinants for the corresponding cyclic boundary problems on the lattices of the periodicity.

Original languageEnglish
Pages (from-to)11-68
Number of pages58
JournalIntegral Equations and Operator Theory
Volume46
Issue number1
DOIs
StatePublished - 2003

Bibliographical note

Funding Information:
Supported by Kamea Project for Scientific Absorption in Israel and partially by a grant from the Israel Science Foundation.

Keywords

  • Boundary problem
  • Essential spectrum
  • Periodic potential
  • Schrödinger operator

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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