Spectral properties of a limit-periodic Schrödinger operator in dimension two

2010/08/27 by Karpeshina, Yulia, Lee, Young-Ran
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1008.4632

Abstract

We study 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 e^{i⟨ k, x⟩} at the high energy region. Second, the isoenergetic curves in the space of momenta 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.

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