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

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 $V_k\in L^{\mathrm{\infty }}(B_3)$ such that $\mathrm{\Delta }{u}_k=V_k{u}_k$, $u_k{\mathrm{\mid }}_{\mathrm{\partial }{B}_3}=0$, ${\mathrm{o}\mathrm{r}\mathrm{d}}_0{u}_k=k$, and $\mathrm{\parallel }{V}_k{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{k}^{3\mathrm{/}2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $c{N}^{2\mathrm{/}3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}^{2\mathrm{/}3})$ 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.