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

The authors consider a Schrödinger operator H=-\Delta +V(\vec x) in dimension two with a quasi-periodic potential V(\vec x). They 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^i\langle \vec \varkappa ,\vec x\rangle in the high energy region. Second, the isoenergetic curves in the space of momenta \vec \varkappa 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 a previous paper on the quasiperiodic polyharmonic operator (-\Delta )^l+V(\vec x), l>1. Here the authors address technical complications arising in the case l=1. However, this text is self-contained and can be read without familiarity with the previous paper.