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

TL;DR

This paper establishes a quantum ergodic restriction theorem linking the ergodic properties of eigenfunctions on a manifold to their Cauchy data on a hypersurface, using Rellich identities to connect local and global quantum ergodicity.

Contribution

It introduces a novel method relating quantum ergodicity of eigenfunctions to their Cauchy data on hypersurfaces via Rellich identities, extending quantum ergodic restriction results.

Findings

Quantum ergodic restriction theorem proved for Cauchy data.

Cauchy data inherits quantum ergodicity if eigenfunctions are globally ergodic.

Cauchy data is quantum uniquely ergodic under global QUE conditions.

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 unique ergodic on the global manifold $M$, then the Cauchy data is automatically quantum unique ergodic on $H$ with respect to operators whose symbols vanish to order one on the glancing set of unit tangential directions to $H$.