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

Published online by Cambridge University Press:  20 November 2018

N. Burq
N. Burq*
Affiliation:
Université Paris Sud Mathématiques, Bât 425, 91405 Orsay Cedex, France e-mail: Nicolas.burq@math.u-psud.fr
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Abstract

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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 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

Résumé

Soit $M$ un domain borné de ${{\mathbb{R}}^{d}}$ 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.

Keywords

35Q5535BXX37K0537L5081Q20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 3 - 15
Copyright
Copyright © Canadian Mathematical Society 2005

References

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