Extended states for the schrodinger operator with quasi-periodic potential in dimension two

Academic Article

Abstract

  • We consider a Schr odinger operator H = -δ + V (x→) in dimension two with a quasi-periodic potential V (x→). We prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e in the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on our previous paper Multiscale analysis in momentum space for quasi-periodic potential in dimension two on the quasiperiodic polyharmonic operator (-δ) + V (x→), l > 1. Here we address technical complications arising in the case l = 1. However, this text is self-contained and can be read without familiarity with our previous work. i l
  • Digital Object Identifier (doi)

    Pubmed Id

  • 14512553
  • Author List

  • Karpeshina Y; Shterenberg R
  • Start Page

  • 1
  • End Page

  • 152
  • Volume

  • 258
  • Issue

  • 1239