On the location of maximal of solutions of Schr\"odinger's equation

We prove an inequality with applications to solutions of the Schr"odinger equation. There is a universal constant $c>0$, such that if $\Omega \subset $\mathbb{R}$^2$issimplyconnected,$u:\Omega \rightarrow $\mathbb{R}$$ vanishes on the boundary $\mathrm{\partial }\mathrm{\Omega }$, and $\mathrm{\mid }u\mathrm{\mid }$ assumes a maximum in $x_0 \in \Omega$,then$$ \inf_{y \in \partial \Omega}{ | x_0 - y|} \geq c \left| \frac{\Delta u}{u} \right|^{-1/2}{L^{\infty}(\Omega)}.$$ It was conjectured by P'olya & Szeg\H{o} (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency $\lambda$ contains a disk of size $\sim \lambda^{-1/2}$. Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance $\sim \lambda^{-1/2}$ from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and Georgiev & Mukherjee): if $u$ solves $-\mathrm{\Delta }u=Vu$ on $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim |V|{L^{\infty}(\Omega)}^{-1/2}$centeredatthepointinwhich$|u|$ assumes a maximum is almost fully contained in $\mathrm{\Omega }$ in the sense that $|B \cap \Omega| \geq 0.99 |B|.$