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

Abstract

We study Schrödinger operator $H=-\Delta +V(x)$ 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 $e^{i⟨\vec{k},\vec{x}⟩}$ 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.

89 pages, 6 figures

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• Periodic and limit-periodic discrete Schrödinger operators