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

TL;DR

This paper demonstrates the existence of extended states with absolutely continuous spectrum for a two-dimensional Schr"odinger operator with a quasi-periodic potential, revealing that high-energy eigenfunctions resemble plane waves with Cantor-like isoenergetic curves.

Contribution

It introduces a novel multiscale analysis method in momentum space to establish the presence of extended states and describes their properties in a quasi-periodic setting for the Schr"odinger operator.

Findings

Absolutely continuous spectrum contains a semi-axis.

Eigenfunctions resemble plane waves at high energies.

Isoenergetic curves are distorted circles with Cantor structure.

Abstract

We consider a Schr\"odinger operator $H=-\Delta+V(\vec x)$ 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 $\vec \varkappa$ 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 $(-\Delta)^l+V(\vec x)$, $l>1$. 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].