Mathematical Research Letters

Volume 20 (2013)

Number 3

Quantum ergodic restriction for Cauchy data: Interior QUE and restricted QUE

Pages: 465 – 475



Hans Christianson (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)

John A. Toth (Department of Mathematics and Statistics, McGill University, Montreal, Canada)

Steve Zelditch (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)


We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich-type identity to relate quantum ergodicity of Cauchy data on $H$ to quantum ergodicity of eigenfunctions on the global manifold $M$. This has the interesting consequence that if the eigenfunctions are quantum uniquely ergodic on the global manifold $M$, then the Cauchy data is automatically quantum uniquely ergodic on $H$ with respect to operators whose symbols vanish to order one on the glancing set of unit tangential directions to $H$.

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