The time-independent Schr\"odinger equation with an isotropic velocity-dependent potential is considered. Treating the velocity-dependent interaction as a small perturbation, we develop analytical formulas for the changes in the scattering phase shifts and wave functions. It is shown that only the zeroth-order solution and the perturbing potential are needed to determine the phase-shift and wave-function corrections. No prior knowledge of the unperturbed scattering-states continuum is required. In order to test the validity of our approach, we applied it to an exactly solvable model for nucleon-nucleon scattering. The results of the perturbation formalism compare quite well with those of the exactly solvable model. The developed formalism can be applied in problems concerning pion-nucleon, nucleon-nucleon, and electron-atom scattering. It may also be useful in studying the scattering of electrons in semiconductor heterostructures.
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M. I. Jaghoub (2006) studied this question.