Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher

Abstract

Let $B_3=B(0,3)\subset\mathbb{R}^3$. For every sufficiently large integer $k$, we construct a nonzero real function $u_k\in C^2(\overline{B_3})$ and a real potential $V_k\in L^\infty(B_3)$ such that $Δu_k=V_ku_k$, $u_k|_{\partial B_3}=0$, $\mathrm{ord}_0u_k=k$, and $\|V_k\|_\infty\le Ck^{3/2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $cN^{2/3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\|V\|_\infty^{2/3})$ is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.