Multiscale analysis in momentum space for quasi-periodic potential in dimension two

Academic Article


  • We consider a polyharmonic operator H=(-δ) +V({combining right arrow above}) in dimension two with l ≥ 2, l being an integer, and a quasi-periodic potential V({combining right arrow above}). 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 at the high energy region. Second, the isoenergetic curves in the space of momenta κ{script}{combining right arrow above} 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. © 2013 AIP Publishing LLC. l i(κ{script}{combining right arrow above},{combining right arrow above})
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    Digital Object Identifier (doi)

    Author List

  • Karpeshina Y; Shterenberg R
  • Volume

  • 54
  • Issue

  • 7