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 | References | Similar Articles | Additional Information

Abstract: We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and 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 as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .

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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:
https://doi.org/10.1090/S0002-9939-06-08597-2

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.