Solution
= Solution
The Gaussian transform is
$$
\widetilde V(\mathbf q)=(2\pi)^{3/2}V_0r_0^3e^{-q^2r_0^2/2}.
$$
Therefore
$$
\frac{d\sigma}{d\Omega}
=\frac{2\pi m^2V_0^2r_0^6}{\hbar^4}e^{-q^2r_0^2},
\qquad q=2k\sin(\vartheta/2).
$$
A sufficient weak-scattering regime is <kinetic energy> much larger than $|V_0|$ (more intrinsically $m|V_0|r_0^2/\hbar^2\ll1$).