On the Location of Maxima of Solutions of Schrödinger's Equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant c > 0 such that if is simply connected, vanishes on the boundary ?Ω, and |u| assumes a maximum in , then (1) It was conjectured by Pólya and Szeg? (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency λ contains a disk of size . Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and of Georgiev and Mukherjee): if u solves on with Dirichlet boundary conditions, then the ball B with radius centered at the point in which |u| assumes a maximum is almost fully contained in Ω in the sense that © 2018 Wiley Periodicals, Inc.