Peierls' substitution for low lying spectral energy windows

TL;DR

This paper extends the analysis of spectral structures in 2D magnetic Schrödinger operators under weak, slowly varying magnetic fields, addressing cases with non-zero Chern numbers and eigenvalue crossings.

Contribution

It generalizes previous results by considering scenarios where the lowest Bloch eigenvalue has a non-zero Chern number or crosses with another eigenvalue.

Findings

Spectral islands form at the bottom of the spectrum under new conditions.

The analysis includes cases with non-zero Chern numbers.

Eigenvalue crossings are incorporated into the spectral structure understanding.

Abstract

We consider a $2d$ magnetic Schr\"odinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a `Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.