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

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⟨ϰ,x⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta ϰ 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.