Quantum ergodicity for contact metric structures

TL;DR

This paper proves a Quantum Ergodicity theorem for eigenfunctions of subLaplacians on contact metric manifolds, assuming ergodic Reeb flow, using semiclassical pseudodifferential calculus and microlocal projectors.

Contribution

It introduces a semiclassical pseudodifferential calculus for contact manifolds and constructs Landau projectors to establish quantum ergodicity for subLaplacians.

Findings

Quantum ergodicity holds for eigenfunctions under ergodic Reeb flow.

Development of a specialized pseudodifferential calculus for contact manifolds.

Construction of microlocal Landau projectors that commute with the subLaplacian.

Abstract

This paper is dedicated to the proof of a Quantum Ergodicity (QE) theorem for the eigenfunctions of subLaplacians on contact metric manifolds, under the assumption that the Reeb flow is ergodic. To do so, we rely on a semiclassical pseudodifferential calculus developed for general filtered manifolds that we specialize to the setting of contact manifolds. Our strategy is then reminiscent of an implementation of the Born-Oppenheimer approximation as we rely on the construction of microlocal projectors in our calculus which commute with the subLaplacian, called Landau projectors. The subLaplacian is then shown to act effectively on the range of each Landau projector as the Reeb vector field does. The remainder of the proof follows the classical path towards QE, once microlocal Weyl laws have been established.