Padé Approximant in Potential Scattering

Abstract

It is suggested that the Pad\'e approximant be used as an approximation outside the radius of convergence of the Born series for scattering from a short-range potential free from a strong singularity. Following Weinberg's analysis of the Born series, one approximates the potential by a finite number of separable potentials in order to deal with a limited range of energy, and thus the associated $K$ matrix can be evaluated without further approximation. When a number $N$ of separable potentials are constructed by using the lowest $2N$ terms of the Born series, the result is identical to the ($N, N$) Pad\'e approximant. If expanded formally into a power series in the potential strength, this approximation reproduces the original Born series exactly up to the $2N\mathrm{th}$ order; but without an expansion it takes a closed form for any finite potential strength, and therefore it is well defined outside the radius of convergence of the Born series. When the Weinberg eigenvalues and eigenfunctions on the energy shell are closely approximated with a suitably chosen $N$, the divergence difficulty of the conventional Born series can be overcome by the use of the Pad\'e approximant, because the latter is correctly continued analytically outside the convergence radius of its Born series.