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

Abstract

We prove an inequality with applications to solutions of the Schr\"odinger equation. There is a universal constant $c>0$, such that if $\Omega \subset \mathbb{R}^2$ is simply connected, $u:\Omega \rightarrow \mathbb{R}$ vanishes on the boundary $\partial \Omega$, and $|u|$ 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 $-\Delta u = Vu$ on $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim \|V\|_{L^{\infty}(\Omega)}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $\Omega$ in the sense that $|B \cap \Omega| \geq 0.99 |B|.$