Isolated singularities of the solutions of the Schrödinger equation with a radial potential

Abstract

We study the behavior near an isolated singularity, say x=0, of solutions of the equation (1) Δu(x)=V(x) u(x), defined in a neighborhood of 0, in terms of the potential V which we assume to be radially symmetric. For a class of potentials P1 that we characterize completely there is a unique singularity with constant sign, behaving like the fundamental solution of the Laplace operator. For a larger class P2 we obtain a unique type of nonnegative singularity that is radially symmetric. If V(r) = crθ, with c>0, θ e R, the class P1 is characterized by θ>-2 and the class P2 by θ≧-2. The results fail for θ<-2.