On the Convergence of the Born Series for All Energies

Formal solution of the Schrödinger equation for nonrelativistic scattering by a spherically symmetric static potential −μV(r) leads to a power series in the real parameter μ for the scattering amplitude (the Born series). It is shown that if ${\int }_0^{\infty }{\mathrm{r|V(r)|dr<\infty ,\hspace{0.17em}\int }}_0^{\infty }{\mathrm{\hspace{0.17em}r}}^2\mathrm{|V(r)|dr<\infty }$ and if −μ|V(r)| is too weak to support a bound state, then the Born series converges at all energies. The method gives a lower bound for the radius of convergence of the Born series which is exact if V ⩾ 0.