# Bethe-Sommerfeld conjecture in semiclassical settings

## TL;DR

This paper proves that under certain conditions, the spectrum of a perturbed elliptic operator in semiclassical settings covers specific intervals, extending the Bethe-Sommerfeld conjecture to these cases.

## Contribution

It establishes the Bethe-Sommerfeld conjecture for a class of semiclassical elliptic operators with periodic perturbations, including generalizations.

## Key findings

- Spectrum covers the interval $( au- ext{small}, au+ ext{small})$
- Results hold for operators with periodic perturbations in semiclassical regime
- Extends the Bethe-Sommerfeld conjecture to new operator classes

## Abstract

Under certain assumptions (including $d\ge 2)$ we prove that the spectrum of a scalar operator in $\mathscr{L}^2(\mathbb{R}^d)$ \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} covers interval $(\tau-\epsilon,\tau+\epsilon)$, where $A^0$ is an elliptic operator and $B(x,hD)$ is a periodic perturbation, $\varepsilon=O(h^\varkappa)$, $\varkappa>0$. Further, we consider generalizations.