Modified effective-range function

We derive a general formula for a modified effective-range function (MERF), $K_l^M\left(k^2\right)$, for all partial waves, $l=0,1,\dots$. This is a generalization of the effective-range function associated with a short-range potential, $K_l\left(k^2\right)=k^{2l+1}cot{\delta }_l\mathrm{}\left(k\right)\mathrm{}$. Here $k^2$ is the energy variable and ${\delta }_l\mathrm{}\left(k\right)\mathrm{}$ the phase shift. The MERF $K_l^M\left(k^2\right)$ can be associated with a potential $V\left(r\right)\mathrm{}$ that allows a decomposition into a long-range and a short-range component. It is a complex real-meromorphic function of $k^2$ in the complex $k$ plane in a domain containing the origin. This (large) domain is determined by the short-range part of the potential. We give a simple formula for $K_l^M\left(k^2\right)$, valid for all $l=0,1,\dots$. It can be used if the long-range part of the potential is analytic at $r=0\mathrm{}$. For $l=0\mathrm{}$ we have the simple expression $K_0^M\left(k^2\right)={\left|f_0\mathrm{}\right(k\left)\right|}^{-2\mathrm{}}k\left[cot{\delta }_0^M\mathrm{}\right(k)-i]+\frac{f_0^{\prime }\mathrm{}(k,0\mathrm{})\mathrm{}}{f_0\mathrm{}\left(k\right)\mathrm{}}.$ Here $f_0\mathrm{}\left(k\right)\mathrm{}$ and $f_0\mathrm{}(k,r)\mathrm{}$ are the Jost function and Jost solution, respectively, associated with the long-range part of the potential, and ${\delta }_0^M\mathrm{}\left(k\right)\mathrm{}$ is the difference between the $s$-wave phase shift associated with the total potential and that of the long-range potential. The prime in $f_0^{\prime }\mathrm{}(k,0\mathrm{})\mathrm{}$ denotes differentiation with respect to $r$. The extension to the case of the Coulomb potential which violates the condition of analyticity at $r=0\mathrm{}$ is briefly discussed.