Contour and surface integrals in potential scattering

Abstract: When the Schr\"{o}dinger equation for stationary states is studied for a system described by a central potential in $n$-dimensional Euclidean space, the radial part of stationary states is an even function of a parameter $\lambda$ which is a linear combination of angular momentum quantum number $l$ and dimension $n$, i.e., $\lambda=l+{(n-2)\over 2}$. Thus, without setting a priori $n=3$, complex values of $\lambda$ can be achieved, in particular, by keeping $l$ real and complexifying $n$. For suitable values of such an auxiliary complexified dimension, it is therefore possible to obtain results on scattering amplitude and phase shift that are completely equivalent to the results obtained in the sixties for Yukawian potentials in $\mathbb{R}^3$. Moreover, if both $l$ and $n$ are complexified, the possibility arises of recovering the partial wave amplitude from residues of a function of two complex variables. Thus, the complex angular momentum formalism can be imbedded into a broader framework, where a correspondence exists between the scattering amplitude and a skew curve in $\mathbb{R}^3$.