Quantum ergodicity for products of hyperbolic planes

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April 2008, 2(2): 287-313. doi: 10.3934/jmd.2008.2.287

Quantum ergodicity for products of hyperbolic planes

1.	Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received August 2007 Revised November 2007 Published January 2008

Abstract
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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: hyperbolic plane., quantum ergodicity.

Mathematics Subject Classification: Primary: 81Q50, Secondary: 43A8.

Citation: Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287

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