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