Bethe-Sommerfeld Conjecture

TL;DR

This paper proves that for a Schrödinger operator with smooth periodic potential in multiple dimensions, only finitely many spectral gaps exist, advancing understanding of spectral properties in quantum mechanics.

Contribution

It establishes the finiteness of spectral gaps for Schrödinger operators with smooth periodic potentials in dimensions two and higher, a significant theoretical result.

Findings

Finitely many spectral gaps in the operator's spectrum.

Extension of spectral gap results to higher dimensions.

Supports the Bethe-Sommerfeld conjecture.

Abstract

We consider Schroedinger operator $-\Delta+V$ in $R^d$ ($d\ge 2$) with smooth periodic potential $V$ and prove that there are only finitely many gaps in its spectrum.