Spreading of Quasimodes in the Bunimovich Stadium

We consider Dirichlet eigenfunctions uλ of the Bunimovich stadium S, satisfying (∆− λ2)uλ = 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 → ∞. We obtain a lower bound Cλ on the L mass of uλ in W , assuming that uλ itself is L -normalized; in other words, the L norm of uλ is controlled by λ 2 times the L norm in W . Moreover, if uλ is a o(λ ) quasimode, the same result holds, while for a o(1) quasimode we prove that L norm of uλ is controlled by λ 4 times the L norm in W . We also show that the L 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 norm of uλ is controlled by the L 2 norm in any pair of strips contained in R, but adjacent to W .