Abstract
This is an an nouncement of the following results. We consider the Schrödinger operator H=−Δ+V(x) in dimension two, V(x) being a limitperiodic potential. We prove that the 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{ei257-1} 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). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.
Research partially supported by USNSF Grant DMS-0800949.
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Dedicated to Boris S. Pavlov on the occasion of his 70th birthday.
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Karpeshina, Y., Lee, YR. (2008). On the Schrödinger Operator with Limit-periodic Potential in Dimension Two. In: Janas, J., Kurasov, P., Naboko, S., Laptev, A., Stolz, G. (eds) Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, vol 186. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8755-6_13
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