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

TL;DR

This paper establishes a universal inequality relating the location of maximum points of solutions to Schrödinger's equation to the potential's magnitude, refining understanding of eigenfunction localization and boundary behavior.

Contribution

It proves a new inequality linking maximum point location to the potential, extending previous results and providing a refined geometric understanding of solutions.

Findings

Maximum points are at a distance proportional to the inverse square root of the potential.

The maximum amplitude point is near the boundary within a scale of the eigenvalue.

In higher dimensions, the maximum point is contained within a large portion of a ball of radius related to the potential.

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 $|B \cap \Omega| \geq 0.99 |B|.$