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On the location of maximal 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 $Ω\subset \mathbb{R}^2$ is simply connected, $u:Ω\rightarrow \mathbb{R}$ vanishes on the boundary $\partial Ω$, and $|u|$ assumes a maximum in $x_0 \in Ω$, then $$ \inf_{y \in \partial Ω}{ \| x_0 - y\|} \geq c \left\| \frac{Δu}{u} \right\|^{-1/2}_{L^{\infty}(Ω)}.$$ It was conjectured by Pólya \& Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency $λ$ contains a disk of size $\sim λ^{-1/2}$. Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance $\sim λ^{-1/2}$ from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and Georgiev \& Mukherjee): if $u$ solves $-Δu = Vu$ on $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim \|V\|_{L^{\infty}(Ω)}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $Ω$ in the sense that $|B \cap Ω| \geq 0.99 |B|.$