Quantum unique ergodicity for magnetic Laplacians on T^2

TL;DR

This paper proves quantum unique ergodicity for magnetic Laplacians on a 2-torus under certain conditions, showing high energy eigenfunctions become uniformly distributed despite non-ergodic classical flows.

Contribution

It establishes quantum unique ergodicity for magnetic Schrödinger operators on the 2-torus with a specific geometric magnetic field condition, even when classical flow is not ergodic.

Findings

High energy eigenfunctions are uniformly distributed.

Quantum unique ergodicity holds under a geometric magnetic field condition.

Results apply despite non-ergodic classical flows.

Abstract

Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schr\"odinger operator. Under a geometric condition on the magnetic field, we show that every sequence of high energy eigenfunctions satisfies the quantum unique ergodicity property even if the Liouville measure is not ergodic for the underlying classical flow (the Euclidean geodesic flow on the 2-torus).