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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Spreading of quasimodes in the Bunimovich stadium
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by Nicolas Burq, Andrew Hassell and Jared Wunsch PDF
Proc. Amer. Math. Soc. 135 (2007), 1029-1037 Request permission


We consider Dirichlet eigenfunctions $u_\lambda$ of the Bunimovich stadium $S$, satisfying $(\Delta - \lambda ^2) u_\lambda = 0$. Write $S = R \cup W$ where $R$ is the central rectangle and $W$ denotes the “wings,” i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in $R$ as $\lambda \to \infty$. We obtain a lower bound $C \lambda ^{-2}$ on the $L^2$ mass of $u_\lambda$ in $W$, assuming that $u_\lambda$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_\lambda$ is controlled by $\lambda ^2$ times the $L^2$ norm in $W$. Moreover, if $u_\lambda$ is an $o(\lambda ^{-2})$ quasimode, the same result holds, while for an $o(1)$ quasimode we prove that the $L^2$ norm of $u_\lambda$ is controlled by $\lambda ^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_\lambda$ may be controlled by the integral of $w |\partial _N u|^2$ along $\partial S \cap W$, where $w$ is a smooth factor on $W$ vanishing at $R \cap W$. These results complement recent work of Burq-Zworski which shows that the $L^2$ norm of $u_\lambda$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.
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Additional Information
  • Nicolas Burq
  • Affiliation: Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex France and Institut Universitaire de France
  • MR Author ID: 315457
  • Email:
  • Andrew Hassell
  • Affiliation: Department of Mathematics, Australian National University, Canberra 0200 ACT, Australia
  • MR Author ID: 332964
  • Email:
  • Jared Wunsch
  • Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
  • Email:
  • Received by editor(s): July 18, 2005
  • Received by editor(s) in revised form: October 21, 2005
  • Published electronically: August 31, 2006
  • Additional Notes: This research was partially supported by a Discovery Grant from the Australian Research Council for the second author, and by National Science Foundation grants DMS-0323021 and DMS-0401323 for the third author. The first and third authors gratefully acknowledge the hospitality of the Mathematical Sciences Institute of the Australian National University. The authors thank an anonymous referee for helpful comments.
  • Communicated by: Mikhail Shubin
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 1029-1037
  • MSC (2000): Primary 35Pxx, 58Jxx
  • DOI:
  • MathSciNet review: 2262903