Spectral properties of a limit-periodic Schr\"odinger operator in dimension two

TL;DR

This paper investigates the spectral characteristics of a two-dimensional Schrödinger operator with a limit-periodic potential, revealing a semiaxis spectrum with eigenfunctions resembling plane waves and a Cantor-like structure in momentum space.

Contribution

It demonstrates the existence of an absolutely continuous spectrum and describes the geometric structure of eigenfunctions and isoenergetic curves for the operator.

Findings

Spectrum contains a semiaxis with absolutely continuous spectrum.

Eigenfunctions resemble plane waves at high energy.

Isoenergetic curves are distorted circles with holes, forming a Cantor set.

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 $e^{i\langle \vec k,\vec x\rangle }$ 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.