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Spectral Properties of Schroedinger Operator with a Quasi-periodic Potential in Dimension Two

Wednesday 8th April 2015 - 13:30 to 14:30
INI Seminar Room 1
Co-author: Roman Shterenberg (UAB)

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. A method of multiscale analysis in the momentum space is developed to prove the results.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons