Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

John A. Toth, Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if H ⊂ M is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions φj{pipe}H of an orthonormal basis {φj} of Δ-eigenfunctions of (M, g) to H are quantum ergodic on H. The condition on H is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface. A key step in the proof is that matrix elements 〈 Fφj, φj〉 of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.

Original languageEnglish (US)
Pages (from-to)715-775
Number of pages61
JournalGeometric and Functional Analysis
Issue number2
StatePublished - Apr 2013

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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