JournalsjstVol. 4, No. 1pp. 65–85

Quantum ergodicity for a class of mixed systems

  • Jeffrey Galkowski

    University of California Berkeley, USA
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Abstract

We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrödinger operators with mixed assumptions on the Hamiltonian flow.) Specifically, we assume that the billiard flow has an invariant ergodic component, UU, and study defect measures, μ\mu, of positive density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence satisfies μU=cμLU\mu|_{U}=c\mu_L|_{U}, where μL\mu_L is the Liouville measure. This proves part of a conjecture of Percival [18].

Cite this article

Jeffrey Galkowski, Quantum ergodicity for a class of mixed systems. J. Spectr. Theory 4 (2014), no. 1, pp. 65–85

DOI 10.4171/JST/62