UMR 5584

# 6 juin 2017: 1 événement

### Mardi 6 juin 2017 16:15-17:15 - Yulia Karpeshina - University of Alabama at Birmingham

Séminaire Math-Phys : Spectral and Transport Properties of Schroedinger Operator with a Quasi-periodic Potential in Dimension Two

Résumé : We consider $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a quasi-periodic potential. We prove that the spectrum of $H$contains a semiaxis (Bethe-Sommerfeld conjecture) and that 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 $e^i\langle \vec k,\vec x\rangle$ at the high energy region.
Second, the isoenergetic curves in the space of momenta $\vec k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure).
It is shown that the spectrum corresponding to these eigenfunctions is absolutely continuous. We prove existence of ballistic transport. A method of multiscale analysis in the momentum space is developed to prove the results.

Lieu : Salle A318

En savoir plus : Séminaires Math-Physique