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arXiv · 2510.16212

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

N. Kaiser
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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.

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N. Kaiser. 2025-10-17. Born series for s-wave scattering length and some exact results. https://arxiv.org/abs/2510.16212

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