2010/05/31 by John Toth, Steve Zelditch · 1 citation
Mathematics · #math.AP
published as Annales Henri Poincaré: Volume 13, Issue 4 (2012), Page 599-670 · 62 pages. First in a series
arxiv created 2011/04/22 · arxiv updated 2012/05/02
Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions ϕj |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 (ϕj|H, ∂ ϕj|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.