Born series for s-wave scattering length and some exact results

TL;DR

The work develops a systematic Born-series framework for the $s$-wave scattering length $a_0$ of short-range central potentials, writing $a_0 = A_0(g)\,a$ with $A_0(g)$ expanded in powers of the dimensionless coupling $g$ and coefficients determined from the profile $f$ and the induced potential $\Phi$. It applies this to a wide set of exponentially decaying and truncated potentials, obtaining sixth-order Born approximations and, in many cases, exact closed-form expressions for $A_0(g)$ in terms of special functions (Bessel, modified Bessel, hypergeometric, and other classical functions); the analytic forms reveal the location of bound-state thresholds as poles of $A_0(g)$ (e.g. zeros of $J_0$ or tangent/hyperbolic expressions depending on the profile). The results provide a detailed map between potential shape, low-energy scattering length, and bound-state spectra, including explicit threshold values and convergence radii, along with several solvable examples that serve as benchmarks for the interplay between Born-series coefficients and spectral properties. This contributes to a clearer understanding of low-energy scattering in short-range interactions and offers exact solvable cases useful for testing numerical methods and approximations in quantum scattering theory.

Abstract

In these notes the Born series for the $s$-wave scattering $a_0$ is calculated for a class of central potentials $V(r)$ up to sixth order in a dimensionless coupling strength $g$. Examples of exponentially decaying potentials as well truncated potentials involving a single length-scale $a$ are considered. In certain favorable cases the exact result for the $g$-dependent $s$-wave scattering length $a_0=A_0(g) a$ can be given in terms of special functions. The poles of $A_0(g)$ at increasing positive values of $g$ correspond to the thresholds, where $s$-wave bound-states occur successively. A scattering problem, where $A_0(g)$ is solvable in terms of elementary functions, is also presented.

Figures (12)

• Figure 1: Scattering length $A_0=a_0/a$ versus $g$ for exponential potential $V(r)\sim - g\, e^{-r/a}$.
• Figure 2: $A_0$ as a function of $g$ for spherical well potential.
• Figure 3: $A_0$ as a function of $g$ for truncated $1/r^2$-potential, restricted to range $g\leq 1/4$.
• Figure 4: $A_0$ as a function of $g$ for truncated Coulomb-potential.
• Figure 5: $A_0$ as a function of $g$ for difference of truncated $1/r^2$- and Coulomb-potential.