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# Mathematical Research Letters

## Volume 20 (2013)

### Number 3

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

Pages: 465 – 475

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n3.a5

#### Authors

#### Abstract

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 $H$.