Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators

TL;DR

This paper develops a perturbation theory demonstrating that small periodic perturbations can make spectral gap edges of 2D periodic Schr"odinger operators non-degenerate, with implications for spectral analysis and lattice structure.

Contribution

It introduces a method to ensure non-degenerate spectral gap edges via small periodic perturbations, addressing degeneracy issues in spectral gaps of 2D Schr"odinger operators.

Findings

Perturbations can make all spectral gap edges non-degenerate.

Non-degeneracy is achieved at finitely many points by a single band function.

Changing the lattice of periods may be necessary for non-degeneracy.

Abstract

We consider a two-dimensional periodic Schr\"odinger operator $H=-\Delta+W$ with $\Gamma$ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of $H$. We show that under arbitrary small perturbation $V$ periodic with respect to $N\Gamma$ where $N=N(W)$ is some integer, all edges of the gaps in the spectrum of $H+V$ which are perturbation of the gaps of $H$ become non-degenerate, i.e. are attained at finitely many points by one band function only and have non-degenerate quadratic minimum/maximum. We also discuss this problem in the discrete setting and show that changing the lattice of periods may indeed be unavoidable to achieve the non-degeneracy.