  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 $\mathbb{R}^3$.
