Extended States for the Schrödinger Operator With Quasi-Periodic Potential in Dimension Two

Extended States for the Schrödinger Operator With Quasi-Periodic Potential in Dimension Two Synopsis

The authors consider a Schrodinger 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.

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