On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schroedinger operators

TL;DR

This paper proves that the edges of spectral bands in certain magnetic operators change in a Lipschitz continuous manner as the magnetic field varies, extending previous results and supporting a recent conjecture.

Contribution

It generalizes Bellissard's 1994 result to a broader class of operators and provides evidence for Nenciu's conjecture regarding spectral band behavior.

Findings

Spectral band edges are Lipschitz continuous with respect to magnetic field strength.

The results apply to both discrete and continuous magnetic Schrödinger operators.

Supports the conjecture that spectral bands exhibit Lipschitz continuity in magnetic field variations.

Abstract

We show for a large class of discrete Harper-like and continuous magnetic Schrodinger operators that their band edges are Lipschitz continuous with respect to the intensity of the external constant magnetic field. We generalize a result obtained by J. Bellissard in 1994, and give examples in favor of a recent conjecture of G. Nenciu.