On resonant scattering by a negative potential

The dependence on a coupling constant $g>0$ of the total crosssection $\sigma(g)$ (of a fixed energy of a particle) for quantum scattering by a potential $gV(x)$ with a compact support is considered. It is found that for a central potential with a nontrivial negative part $\sigma(g)$ is unbounded for some sequence $g_l\to\infty$. Moreover the lower estimate $\sigma(g_l)\geqslant cg_l^{1/2}$, $c>0$, holds. On the contrary, for a positive repulsive potential (not necessarily central) the total cross-section is shown to be bounded uniformly in $g$.