Spreading of quasimodes in the Bunimovich stadium

Abstract

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