On the rate of quantum ergodicity I: Upper bounds

Steven Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

Abstract

One problem in quantum ergodicity is to estimate the rate of decay of the sums {Mathematical expression} on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λj, φ{symbol}j} are the spectral data of the Δ of (M, g), A is a 0-th order ψDO, {Mathematical expression} is the (Liouville) average of its principal symbol and {Mathematical expression}. That Sk(λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show that Sk(λ;A)=O((log λ)-k/2) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).

Original languageEnglish (US)
Pages (from-to)81-92
Number of pages12
JournalCommunications in Mathematical Physics
Volume160
Issue number1
DOIs
StatePublished - Feb 1 1994

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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