Eigenvalues in spectral gaps of differential operators

Marco Marletta, Rob Scheichl

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

Spectral problems with band-gap spectra arise in numerous applications, including the study of crystalline structures and the determination of transmitted frequencies in photonic waveguides. Numerical discretization of these problems can yield spurious results, a phenomenon known as spectral pollution. We present a method for calculating eigenvalues in the gaps of self-adjoint operators which avoids spectral pollution. The method perturbs the problem into a dissipative problem in which the eigenvalues to be calculated are lifted out of the convex hull of the essential spectrum, away from the spectral pollution. The method is analysed here in the context of one-dimensional Schrödinger equations on the half line, but is applicable in a much wider variety of contexts, including PDEs, block operator matrices, multiplication operators, and others.

Original languageEnglish
Pages (from-to)293-320
Number of pages28
JournalJournal of Spectral Theory
Volume2
Issue number3
DOIs
Publication statusPublished - 2012

Keywords

  • Discretization
  • Dissipative
  • Eigenvalue
  • Essential spectrum
  • Schrödinger
  • Self-adjoint
  • Spectral band
  • Spectral gap
  • Spectral pollution
  • Variational method

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Geometry and Topology

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