Journal of Modern Dynamics (JMD)

Quantum ergodicity for products of hyperbolic planes

Pages: 287 - 313, Issue 2, April 2008      doi:10.3934/jmd.2008.2.287

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Dubi Kelmer - Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email)

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 cannot be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.

Keywords:  quantum ergodicity, hyperbolic plane.
Mathematics Subject Classification:  Primary: 81Q50, Secondary: 43A85.

Received: August 2007;      Revised: November 2007;      Available Online: January 2008.