### Abstract

We consider Dirichlet eigenfunctions μλ of the Bunimovich stadium S, satisfying (δ-λ^{2})μλ = 0. Write S = R.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 λ → εWe obtain a lower bound Cλ^{-2} on the L^{2} mass of uλ in W, assuming that uλ itself is L^{2}-normalized; in other words, the L^{2} norm of uλ is controlled by λ^{2} times the L^{2} norm in W. Moreover, if uλ is an o(λ.2) quasimode, the same result holds, while for an o(1) quasimode we prove that the L^{2} norm of uλ is controlled by λ^{4} times the L^{2} norm in W. We also show that the L^{2} norm of uλ may be controlled by the integral of w|.Nu| ^{2} along.S.W, where w is a smooth factor on W vanishing at R.W. These results complement recent work of Burq-Zworski which shows that the L ^{2} norm of uλ is controlled by the L^{2} norm in any pair of strips contained in R, but adjacent to W.

Original language | English (US) |
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Pages (from-to) | 1029-1037 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 135 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2007 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*135*(4), 1029-1037. https://doi.org/10.1090/S0002-9939-06-08597-2