Weyl's law and quantum ergodicity for maps with divided phase space

Jens Marklof*, Stephen O'Keefe, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdière theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our main theorems to quantized linked twist maps on the torus. In the appendix, Zelditch connects these studies to some earlier results on 'pimpled spheres' in the setting of Riemannian manifolds. The common feature is a divided phase space with a periodic component.

Original languageEnglish (US)
Pages (from-to)277-304
Number of pages28
JournalNonlinearity
Volume18
Issue number1
DOIs
StatePublished - Jan 2005

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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