Yulia Karpeshina, Young-Ran Lee
Spectral properties of a limit-periodic Schr\"{o}dinger
operator in dimension two
(549K, LaTeX 2e with 5 EPS Figures)
ABSTRACT. 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 $e^{i\langle ec k,ec
x
angle }$ at the high energy region. Second, the isoenergetic
curves in the space of momenta $ec k$ 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.