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

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

2010-08-27

arxiv.org/abs/1008.4632v1

Physics

Mathematical Physics

89 pages, 6 figures

Scientific paper

We study Schr\"{o}dinger 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 $e^{i\langle \vec k,\vec x\rangle }$ 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.

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