Quantum Ergodicity for products of hyperbolic planes

TL;DR

This paper extends quantum ergodicity results to products of hyperbolic planes, demonstrating that eigenfunctions become equidistributed on specific submanifolds where the geodesic flow is ergodic.

Contribution

It proves quantum ergodicity on submanifolds of products of hyperbolic planes, where the geodesic flow is ergodic, despite non-ergodicity on the entire space.

Findings

Almost all eigenfunctions are equidistributed on ergodic submanifolds.

The result generalizes quantum ergodicity to non-ergodic settings with constants of motion.

Eigenfunctions concentrate on submanifolds where ergodicity holds.

Abstract

For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half planes, the geodesic flow has constants of motion so it can not be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. In accordance, we show that almost all eigenfunctions become equidistributed on these submanifolds.