Zeroes of Schrödinger eigenfunctions at potential singularities (Q1059775)

The note contains the proof of the following proposition. Let \(\Omega\) be a domain in \({\mathbb{R}}^ N\) (N\(\geq 3)\) and \(u\in C(\Omega)\) be a function satisfying in distribution sense the Schrödinger equation \(-\Delta u+qu=\lambda u\) with \(0\leq q\in L^ 1_{loc}(\Omega),\) \(\lambda\in {\mathbb{R}}\). Then \(u(0)=0\) unless \(\overline{\lim}_{r\downarrow 0}\quad r^{-N+2}\int_{| x| <r}q(x)dx<\infty.\) In particular, if \(q(x)\geq \rho | x|^{-(2+\alpha)}\), \(x\in \Omega\), \(\rho,\alpha >0\) then every continuous solution u vanishes at 0. This proposition is a generalization allowing an a bit stronger singularity of q(x) at the origin then the result by \textit{S. H. Alimov} and \textit{I. Yoo'} [Acta Sci. Math. (to appear)] who proved the same property of u for \(q=q_ 0+q_ 1\geq 0\), \(q_ 0\in {\mathcal L}^ 2(\Omega)\) is a radially symmetric function, \(q_ 0\leq C| x|^{-2}\), \(q_ 1\in {\mathcal L}^{\infty}(\Omega)\).