On absolute continuity of the spectrum of a periodic magnetic Schr\"odinger operator

TL;DR

This paper proves that the spectrum of a periodic magnetic Schr"odinger operator in higher dimensions is absolutely continuous under certain regularity conditions on the electric and magnetic potentials.

Contribution

It establishes absolute continuity of the spectrum for a class of periodic magnetic Schr"odinger operators with specific regularity assumptions on the potentials.

Findings

Spectrum is absolutely continuous under given conditions.

Results apply when electric potential is in L^{n/2}_loc and magnetic potential in H^q_loc.

Provides conditions ensuring spectral properties of magnetic Schr"odinger operators.

Abstract

We consider the Schr\"odinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of the operator in question under some conditions which, in particular, are satisfied if $V\in L^{n/2}_{{\mathrm {loc}}}({\mathbb R}^n)$ and $A\in H^q_{{\mathrm {loc}}}({\mathbb R}^n;{\mathbb R}^n)$, $q>(n-1)/2$.