Quantum Ergodic Restriction Theorems. I: Interior Hypersurfaces in Domains with Ergodic Billiards

John A. Toth, Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions φ j{pipe}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 in a piecewise-analytic Euclidean domain, the Cauchy data (φ j{pipe}H, ∂ H νφ j{pipe}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.

Original languageEnglish (US)
Pages (from-to)599-670
Number of pages72
JournalAnnales Henri Poincare
Volume13
Issue number4
DOIs
StatePublished - May 1 2012

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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