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Ergodic properties of eigenfunctions for the Dirichlet problem. (English) Zbl 0788.35103
The authors generalize to the Dirichlet-Laplacian a result already known for compact Riemannian manifolds and for semiclassical Schrödinger operators: namely that under ergodicity assumptions on the classical flow, there exists a subfamily of density 1 of orthonormalized eigenfunctions of the Laplacian, which are asymptotically equidistributed in phase space as the energy tends to infinity. The proof is based on refined versions of theorems on propagation of singularities for boundary value problems.

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
58J47 Propagation of singularities; initial value problems on manifolds
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