Quantum ergodic restriction theorems, I: interior hypersurfaces in domains with ergodic billiards

TL;DR

This paper establishes conditions under which restrictions of eigenfunctions to hypersurfaces in ergodic billiard domains exhibit quantum ergodicity, introducing new results on boundary traces and operator orthogonality.

Contribution

It proves quantum ergodic restriction theorems for hypersurfaces in ergodic billiard domains, including conditions for individual and joint quantum ergodicity of boundary data.

Findings

Quantum ergodic restriction holds for smooth hypersurfaces with appropriate weighting.

Conditions involving Poincaré maps determine individual quantum ergodicity of boundary data.

New local Weyl law and almost-orthogonality results for Fourier integral operators are established.

Abstract

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(\phi_j|H, \partial \phi_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincar\'e maps for $H$. The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an 'almost-orthogonality' result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow.