Improvement of the Born Series at Low Energy

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Abstract
The Born series for a quantum-mechanical Green function is studied. A prescription is given for making "best" use of the information contained in the first few terms of that series, and, in particular, for calculating bound states or resonances from them. This prescription is based on heuristic convergence arguments whose formal steps are somewhat reminiscent of renormalization group methods. The present considerations may be applied to potential scattering as well as to quantum field theory. They are expected to be valid for low-energy phenomena and finite-range forces. The prescription is tested, using only the first two Born terms, in the case of a nonrelativistic particle moving in a Yukawa potential: For well depths producing a single shallow bound state, the usual effective-range results are closely reproduced, and, in some ways, improved upon.