Shuo-Wen Feng

Vanishing orders of Dirichlet solutions to the Schr\"odinger 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 $V_k\in L^\infty(B_3)$ such that $\Delta u_k=V_ku_k$, $u_k|_{\partial B_3}=0$, $\mathrm{ord}_0u_k=k$, and $\|V_k\|_\infty\le Ck^{3/2}$. Consequently, for...