Extended States for the Schrodinger Operator with Quasi-Periodic Potential in Dimension Two
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-3543-1 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3543-1 (ISBN)
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Considers a Schrodinger operator $H=-/Delta +V(/vec x)$ in dimension two with a quasi-periodic potential $V(/vec x)$. The authors 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.
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.
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.
Yulia Karpeshina, University of Alabama Birmingham, AL. Roman Shterenberg, University of Alabama Birmingham, AL.
Introduction
Preliminary Remarks
Step I
Step II
Step III
Step IV
Induction
Isoenergetic Sets. Generalized Eigenfunctions of $H$
Proof of Absolute Continuity of the Spectrum
Appendices
List of main notations
Bibliography.
| Erscheinungsdatum | 02.03.2019 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 225 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Naturwissenschaften ► Physik / Astronomie | |
| ISBN-10 | 1-4704-3543-8 / 1470435438 |
| ISBN-13 | 978-1-4704-3543-1 / 9781470435431 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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