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

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $ϕ_j |_H$ to $H$ of $Δ$-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 $(ϕ_j|H, \partial ϕ_j|H)$ 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 Poincaré 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.

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