2012/09/30 by Jeffrey Galkowski · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.AP #math.MP #math.SP
paper · pdf · doi:10.4171/jst/62
published as J. Spectr. Theory 4 (2014), no. 1, 65-85 · 16 pages, 2 figures
arxiv created 2014/02/19 · arxiv updated 2014/07/02
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c muL where muL is the Liouville measure. This proves part of a conjecture of Percival.