Extended States for the Schr\"odinger Operator with Quasi-periodic Potential in Dimension Two

We consider a Schr"odinger operator $H=-\mathrm{\Delta }+V(\vec{x})$ in dimension two with a quasi-periodic potential $V(\vec{x})$. 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 $e^{i\langle \vec \varkappa,\vec x\rangle}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec{\mathrm{\varkappa }}$ 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. The result is based on the previous paper [1] on quasiperiodic polyharmonic operator $(-\mathrm{\Delta }{)}^l+V(\vec{x})$, $l>1$. We address here technical complications arising in the case $l=1$. However, this text is self-contained and can be read without familiarity with [1].