Quantum Ergodicity for products of hyperbolic planes

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.