We consider Dirichlet eigenfunctions μλ of the Bunimovich stadium S, satisfying (δ-λ2)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 L2 mass of uλ in W, assuming that uλ itself is L2-normalized; in other words, the L2 norm of uλ is controlled by λ2 times the L2 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 L2 norm of uλ is controlled by λ4 times the L2 norm in W. We also show that the L2 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 L2 norm in any pair of strips contained in R, but adjacent to W.
ASJC Scopus subject areas
- Applied Mathematics