Quantum Ergodicity of Boundary Values of Eigenfunctions: a Control Theory Approach
نویسنده
چکیده
Consider M , a bounded domain in R, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording 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 Gérard, Leichtnam [7] and Hassel, Zelditch [8] obtained under the additional assumption of the convexity of M . Résumé. Soit M un domain borné de R qui est une variété riemanienne à coins. On suppose que le billard défini par le flot géodésique brisé est ergodique. On démontre que les valeurs au bord des fonctions propres du Laplacien (avec des conditions aux limites raisonnables) sont asymptotiquement équidistribuées dans le bord. Ceci généralise des résultats antérieurs de P. Gérard, E. Leichtnam [7] et A. Hassel, S. Zelditch [8] obtenus sous l’hypothèse supplémentaire de convexité géodésique du domaine.
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Quantum Ergodicity of Boundary Values of Eigenfunctions
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