New Scattering Approximation

The spherical-harmonics decomposition of the wave equation for scattering of a plane wave by a potential of arbitrary space dependence is derived from the Green's function formulation. An approximation is then introduced which reduces the scattering problem to the solution of a set of coupled linear equations for the partial-wave amplitudes. The numerical coefficients in these equations involve Clebsch-Gordan coefficients and integrals over all space of a product of a spherical-harmonic component of the potential with a pair of Bessel functions. For central potentials ($U$), the equations decouple, and the phase shifts reduce to ${tan\delta }_l=\frac{-k\int 0^{\infty }{r}^2\mathrm{dr}{j_l}^2\mathrm{}\left(\mathrm{kr}\right)U\left(r\right)\mathrm{}}{1-k\int 0^{\infty }{r}^2\mathrm{dr}{j}_l\mathrm{}\left(\mathrm{kr}\right)\mathrm{}{y}_l\mathrm{}\left(\mathrm{kr}\right)U\left(r\right)\mathrm{}}.$ While the manipulations involved resemble those in the Born approximation (the numerator above is the Born ${\delta }_l$), the concept of the new approximation is quite different and the results can be very dissimilar. Exact and approximate phase shifts are exhibited for various spherical wells and barriers. For repulsive potentials, the approximation works well for short to moderate ranges ($\mathrm{ka}\lesssim 2$) regardless of barrier height. For attractive potentials, comparable results are obtained for the "potential scattering," but the resonances are ignored.