Bethe-Sommerfeld conjecture for pseudodifferential perturbation

TL;DR

This paper proves that for a class of periodic pseudodifferential operators with certain smoothness and order conditions, the spectrum includes a half-line, confirming a version of the Bethe-Sommerfeld conjecture.

Contribution

It establishes the Bethe-Sommerfeld conjecture for a broad class of pseudodifferential operators with smooth symbols and lower-order perturbations.

Findings

Spectrum contains a half-line under given conditions

Validates Bethe-Sommerfeld conjecture for pseudodifferential operators

Extends known results to operators with less regular perturbations

Abstract

We consider a periodic pseudodifferential operator $H=(-\Delta)^l+A$ ($l>0$) in $\R^d$ which satisfies the following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation $A$ has order smaller than $2l-1$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line.