Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

TL;DR

This paper proves that certain periodic pseudo-differential operators with strong perturbations have spectra containing a half-line, confirming the Bethe-Sommerfeld conjecture for a broad class of periodic Schrödinger operators.

Contribution

It establishes the Bethe-Sommerfeld conjecture for periodic operators with strong perturbations, extending previous results to a wider class of operators.

Findings

Spectrum contains a half-line under given conditions

Confirms Bethe-Sommerfeld conjecture for periodic magnetic Schrödinger operators

Applicable in all spatial dimensions

Abstract

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in $\bx$, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.