Spreading of quasimodes in the bunimovich stadium

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.