Extended States for the Schrodinger Operator with Quasi-Periodic Potential in Dimension Two Paperback / softback
Part of the Memoirs of the American Mathematical Society series
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.
- Format: Paperback / softback
- Pages: 139 pages
- Publisher: American Mathematical Society
- Publication Date: 30/05/2019
- Category: Differential calculus & equations
- ISBN: 9781470435431