Abstract
We study the Schrödinger operator H = −Δ + V(x) in dimension two, V(x) being a limit-periodic 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 exp(\(i\left\langle {\overrightarrow k ,\overrightarrow x } \right\rangle \)) at the high energy region. Second, the isoenergetic curves in the space of momenta \(\overrightarrow k \) corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.
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Supported in part by NSF grant DMS-1201048.
Supported by the National Research Foundation of Korea (NRF)-grant 2009-0064945.
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Karpeshina, Y., Lee, YR. Spectral properties of a limit-periodic Schrödinger operator in dimension two. JAMA 120, 1–84 (2013). https://doi.org/10.1007/s11854-013-0014-1
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DOI: https://doi.org/10.1007/s11854-013-0014-1