On the location of maximal of solutions of Schrödinger's equation

Abstract

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant $c>0$, such that if $\Omega \subset {\mathbb{R}}^2$ is simply connected, $u:\Omega \to \mathbb{R}$ vanishes on the boundary $\partial \Omega$, and $|u|$ assumes a maximum in $x_0\in \Omega$, then $$\underset{y\in \partial \Omega }{\inf }\Vert x_0-y\Vert \ge c{\Vert \frac{\Delta u}{u}\Vert }_{L^{\infty }(\Omega )}^{-1/2}.$$ It was conjectured by Pólya \& Szegő (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 $\sim \Vert V{\Vert }_{L^{\infty }(\Omega )}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $\Omega$ in the sense that $|B\cap \Omega |\ge 0.99|B|.$

v2, to appear in Comm. Pure Appl. Math

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