A spectral theory of linear operators on rigged hilbert spaces under analyticity conditions II: Applications to SchrÖdinger operators

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3 Citations (Scopus)

Abstract

A spectral theory of linear operators on a rigged Hilbert space is applied to Schrödinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one-dimensional Schrödinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function D(λ) is constructed so that D(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that D(λ) is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.

Original languageEnglish
Pages (from-to)375-405
Number of pages31
JournalKyushu Journal of Mathematics
Volume72
Issue number2
DOIs
Publication statusPublished - 2018
Externally publishedYes

Keywords

  • Generalized spectrum
  • Resonance pole
  • Rigged hilbert space
  • Schrödinger operator
  • Spectral theory

ASJC Scopus subject areas

  • Mathematics(all)

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