Exact computation of the scattering length

We present an approach for the exact computation of the scattering length. Our numerical solution takes into account the full tail of the potential energy $V(R)$, and is thus highly accurate. Our approach is based on a change of variable $x=1/R$ at a large enough distance R. The short-range propagation of the solution is done as usual (in $R$), and at $R=R_{\mathrm{mid}}$, we switch to the new variable $x$, which allows to compress the infinite tail into a manageable compact interval for $x:\ [0,1]$. This approach is very convenient and can be implemented with almost any existing propagator. We have tested two numerical methods (log-derivative, and an integral-equation based method) against exactly/analytically soluble potentials of Lennard--Jones type. The agreement between the computed and analytical results reaches a relative error of $10^{-12}$ or better.