Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions $u_{\lambda }$ of the Bunimovich stadium $S$, satisfying $(\mathrm{\Delta }-{\lambda }^2)u_{\lambda }=0$. 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 $\lambda \to \mathrm{\infty }$. We obtain a lower bound $C{\lambda }^{-2}$ on the $L^2$ mass of $u_{\lambda }$ in $W$, assuming that $u_{\lambda }$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_{\lambda }$ is controlled by ${\lambda }^2$ times the $L^2$ norm in $W$. Moreover, if $u_{\lambda }$ is a $o({\lambda }^{-2})$ quasimode, the same result holds, while for a $o(1)$ quasimode we prove that $L^2$ norm of $u_{\lambda }$ is controlled by ${\lambda }^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_{\lambda }$ 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 $u_{\lambda }$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.