Non-smoothness of the fundamental solutions for Schrödinger equations with super-quadratic and spherically symmetric potential

We study non-smoothness of the fundamental solution for the Schrödinger equation with a spherically symmetric and super-quadratic potential in the sence that $V(x)\ge C\mathrm{\mid }x{\mathrm{\mid }}^{2+\epsilon }$ at infinity with constants $C>0$ and $\epsilon >0$. More precisely, we show the fundamental solution $E(t,x,y)$ does not belong to $C^1$ as a function of $(t,x,y)$.