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# Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach

Published:2005-03-01
Printed: Mar 2005
• N. Burq
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## Abstract

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of~$M$.
 MSC Classifications: 35Q55 - NLS-like equations (nonlinear Schrodinger) [See also 37K10] 35BXX - unknown classification 35BXX37K05 - Hamiltonian structures, symmetries, variational principles, conservation laws 37L50 - Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems 81Q20 - Semiclassical techniques, including WKB and Maslov methods