Spectral Properties of Schroedinger Operator with a Quasi-periodic Potential in Dimension Two
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About this item
| Description: |
Karpeshina, Y (University of Alabama at Birmingham)
Wednesday 08 April 2015, 13:30-14:30 |
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| Created: | 2015-04-13 10:21 |
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| Collection: | Periodic and Ergodic Spectral Problems |
| Publisher: | Isaac Newton Institute |
| Copyright: | Karpeshina, Y |
| Language: | eng (English) |
| Abstract: | Co-author: Roman Shterenberg (UAB)
We consider H=−Δ+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 ei⟨k⃗ ,x⃗ ⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta k⃗ 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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