Concentration of Bloch eigenstates in the presence of gauge at the semi-classical limit

TL;DR

This paper proves that in the semi-classical limit, Bloch eigenstates in a periodic channel with a constant gauge concentrate near potential maximizers, contrasting with ground states that concentrate near potential minimizers.

Contribution

It establishes a new concentration result for Bloch eigenstates under a gauge in the semi-classical limit, highlighting differences from ground state behavior.

Findings

Eigenstates concentrate near scalar potential maximizers as gauge approaches a critical value from above.

Ground states concentrate near potential minimizers regardless of gauge in the semi-classical limit.

The concentration behavior depends on the gauge's limit relative to a critical value.

Abstract

We prove a concentration result of a Bloch eigenstate in a periodic channel under a constant gauge. In the semi-classical limit $h--> 0$ these eigenstates concentrate near a maximizer of the scalar potential of the associated Schrodinger operator, provided the constant gauge converges to a critical value from above. This is in contrast with the ground states which concentrate for any gauge in this limit near a minimizer of the scalar potential.