Born Approximation and Large-Momentum-Transfer Processes in Potential Scattering

Several problems of nonrelativistic scattering by analytic potentials are studied, where both the energy and the momentum transfer are large. Specifically, we find asymptotically the reflection coefficients from the one-dimensional potentials $V{e}^x{(1+e^x)}^{-2\mathrm{}}$, $V{e}^{\frac{x}{2}}{(1+e^x)}^{-1\mathrm{}}$, and $V\exp (-x^2)$, and the scattering amplitudes for their three-dimensional generalizations $V{e}^r{(1+e^r)}^{-2\mathrm{}}$, $V{e}^{\frac{r}{2}}{(1+e^r)}^{-1\mathrm{}}$, and $V\exp (-r^2)$. The results are compared with these obtained from Born approximation, and it is found that the first Born approximation gives the correct answer asymptotically in the cases $V{e}^{\frac{x}{2}}{(1+e^x)}^{-1\mathrm{}}$ and $V{e}^{\frac{r}{2}}{(1+e^r)}^{-1\mathrm{}}$, but not in the other cases. Conjectures about more general cases are also given.