Almost periodic Schrödinger operators - III. The absolutely continuous spectrum in one dimension

P. Deift, B. Simon

Research output: Contribution to journalArticle

Abstract

We discuss the absolutely continuous spectrum of H=-d2/dx2+V(x) with V almost periodic and its discrete analog (hu)(n)=u(n+1)+u(n-1)+V(n)u(n). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. V in the hull and a.e. E in A, H and h have continuum eigenfunctions, u, with |u| almost periodic. In the discrete case, we prove that |A|≦4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2 kdk/dE≧π-2 in the continuum case and that 2π sinπkdk/dE≧1 in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.

Original languageEnglish (US)
Pages (from-to)389-411
Number of pages23
JournalCommunications In Mathematical Physics
Volume90
Issue number3
DOIs
StatePublished - Sep 1983

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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