Isolated singularities of the Schrödinger equation with a good potential

Abstract:

We study the behaviour near an isolated singularity, say $0$, of nonnegative solutions of the Schrödinger equation $- \Delta u + Vu = 0$ defined in a punctured ball $0 < |x| < R$. We prove that whenever the potential $V$ belongs to the Kato class ${K_n}$ the following alternative, well known in the case of harmonic functions, holds: either $|x{|^{n - 2}}u(x)$ has a positive limit as $|x| \to 0$ or $u$ is continuous at $0$. In the first case $u$ solves the equation $- \Delta u + Vu = a\delta$ in $\{ |x| < R\}$. We discuss the optimality of the class ${K_n}$ and extend the result to solutions $u \ngeq 0$ of $- \Delta u + Vu = f$.