Extended states for polyharmonic operators with quasi-periodic potentials in dimension two

Academic Article


  • We consider a polyharmonic operator H = (- δ)l + V(x) in dimension two with l ≥ 2, l being an integer, and 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 ei[k, x] at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a 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. © 2012 American Institute of Physics.
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    Author List

  • Karpeshina Y; Shterenberg R
  • Volume

  • 53
  • Issue

  • 10