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Extended States for the Schrödinger Operator with Quasi-periodic Potential in Dimension Two
About this Title
Yulia Karpeshina and Roman Shterenberg
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 258, Number 1239
ISBNs: 978-1-4704-3543-1 (print); 978-1-4704-5069-4 (online)
DOI: https://doi.org/10.1090/memo/1239
Published electronically: February 13, 2019
MSC: Primary 35J10, 35P15, 35P20, 81Q05, 81Q10; Secondary 81Q15, 37K55, 47F05
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminary Remarks
- 3. Step I
- 4. Step II
- 5. Step III
- 6. STEP IV
- 7. Induction
- 8. Isoenergetic Sets. Generalized Eigenfunctions of $H$
- 9. Proof of Absolute Continuity of the Spectrum
- 10. Appendices
- 11. List of main notations
Abstract
We consider a Schrödinger 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 $e^{i\langle \vec \varkappa ,\vec x\rangle }$ 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 our previous paper Multiscale analysis in momentum space for quasi-periodic potential in dimension two on the quasiperiodic polyharmonic operator $(-\Delta )^l+V(\vec x)$, $l>1$. 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.
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