Extended States for the Schrödinger Operator with Quasi-periodic Potential in Dimension Two

We consider a Schrodinger operator H = -Delta + V ((x) over right arrow) in dimension two with a quasi-periodic potential V ((x) over right arrow). 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(i <(chi) over right arrow,(x) over right arrow >) in the high energy region. Second, the isoenergetic curves in the space of momenta (chi) over right arrow 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 our previous paper Multiscale analysis in momentum space for quasi-periodic potential in dimension two on the quasiperiodic polyharmonic operator (-Delta)(l) + V ((x) over right arrow), l > 1. Here we address technical complications arising in the case l = 1. However, this text is self-contained and can be read without familiarity with our previous work.