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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spreading of quasimodes in the Bunimovich stadium


Authors: Nicolas Burq, Andrew Hassell and Jared Wunsch
Journal: Proc. Amer. Math. Soc. 135 (2007), 1029-1037
MSC (2000): Primary 35Pxx, 58Jxx
Published electronically: August 31, 2006
MathSciNet review: 2262903
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Abstract: 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 \vert\partial_N u\vert^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
Email: nicolas.burq@math.u-psud.fr

Andrew Hassell
Affiliation: Department of Mathematics, Australian National University, Canberra 0200 ACT, Australia
Email: hassell@maths.anu.edu.au

Jared Wunsch
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: jwunsch@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08597-2
PII: S 0002-9939(06)08597-2
Keywords: Eigenfunctions, quasimodes, stadium, concentration, quantum chaos
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.