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

Abstract

We consider a polyharmonic operator \documentclass[12pt]{minimal}\begin{document}$H=(-\Delta)^l+V({\vec{x}})$\end{document}H=(−Δ)l+V(x⃗) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential \documentclass[12pt]{minimal}\begin{document}$V({\vec{x}})$\end{document}V(x⃗). 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 \documentclass[12pt]{minimal}\begin{document}$e^{i\langle {\vec{\varkappa }},{\vec{x}}\rangle }$\end{document}ei⟨ϰ⃗,x⃗⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta \documentclass[12pt]{minimal}\begin{document}${\vec{\varkappa }}$\end{document}ϰ⃗ 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.