Quantum unique ergodicity for magnetic Laplacians on T^2
Léo Morin, Gabriel Rivière
TL;DR
The paper proves quantum unique ergodicity for magnetic Laplacians on the 2-torus under a geometric control condition B_∞>0, showing high-energy eigenfunctions become equidistributed in phase space with respect to the Liouville measure, even when the classical geodesic flow is integrable. A novel magnetic Weyl quantization on the torus is developed and used alongside semiclassical measures and a two-microlocal analysis to decompose invariant measures along periodic geodesics. The constant-field case yields a quantitative QE via averaging along the long cyclotron flow, while the general variable-field case uses two-microlocalization to exclude mass on periodic orbits, leading to a Lebesgue-in-x and uniform-in-angle limit. The results highlight a mechanism by which magnetic fields enforce quantum ergodicity in settings where classical dynamics alone would not guarantee it, and they extend QE phenomena beyond ergodic geodesic flows. A complementary appendix discusses more general magnetic-field behavior when B_∞ is not everywhere positive.
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 Schrödinger 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).