Mathematical Research Letters
Volume 20 (2013)
Quantum ergodic restriction for Cauchy data: Interior QUE and restricted QUE
Pages: 465 – 475
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 $H$.