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

  • N. Burq (a1)
Abstract

Consider M, a bounded domain in 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 eigen-functions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of M.

Résumé

Soit M un domain borné de ℝ d 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 et E. Leichtnamaussi bien que A. Hassell et S. Zelditch, obtenus sous l’hypothèse supplémentaire de convexité géodésique du domaine.

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References
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[1] Bardos, C., Lebeau, G., and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. Siam J. Control Optim. 305(1992), 10241065.
[2] Burq, N. and Gérard, P., Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325(1997), 749752.
[3] Burq, N. and Lebeau, G.. Mesures de défaut de compacité, application au système de Lamé. Ann. Sci. École Norm. Sup. 34(2001), 817870.
[4] Burq, N.. Mesures semi-classiques et mesures de défaut. Astérisque 245(1997) 167195.
[5] Chazarain, Jacques and Piriou, Alain. Introduction to the theory of linear partial differential equations. Studies in Mathematics and its Applications, 14, North-Holland, Amsterdam, 1982.
[6] de Verdière, Y. Colin. Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102(1985), 187214.
[7] Gérard, P. and Leichtnam, E.. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(1993), 559607.
[8] Hassell, A. and Zelditch, S.. Quantum ergodicity of boundary values of eigenfunctions. Preprint, 2002.
[9] Lions, P. L. and Paul, T.. Sur les mesures de Wigner. Rev.Mat. Iberoamerican. 9(1993), 553618.
[10] Shnirelman, A. I.. Ergodic properties of eigenfunctions. Uspekhi Mat. Nauk. 29(1974), 181182.
[11] Zelditch, S.. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(1987), 919941.
[12] Zelditch, S. and Zworski, M.. Ergodicity of eigenfunctions for ergodic billiards. Comm. Math. Phy. 175(1996), 673682.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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