Born series for potential scattering

The Born series for the T matrix is considered with the class of potentials V(r)=V00dtA(t)×e-tr2. An explicit knowledge of A(t) is not necessary since the resulting scattering amplitude can be expressed in terms of V. The exact expressions for the first and second Born approximations are utilized to approximate the general nth-order term in the series. The series is explicitly summed to yield an expression for the scattering amplitude which reduces at high energy to the previous impact-parameter amplitude of Blankenbecler and Goldberger. The continuation of the series beyond its radius of convergence is discussed and exemplified with an exponential potential V=V0e-r. A few specific numerical examples are considered to illustrate the behavior of the resulting scattering cross section. Improvement of low-energy total cross sections is noted, particularly for repulsive or weakly attractive potentials.