Short-range interaction in three-dimensional quantum mechanics

We show that it is possible to define shape-independent three-dimensional short-range quantum interactions in two-parameter form for non-spherical angular momentum channels through double rescaling of potential strength. Unlike the special case of l=0, where the zero-range limit of the system is renormalizable, the effective ranges diverge for l≠0 channels, and the system becomes trivial at zero-size limit. It is also shown that the two-parameter representation with finite interaction range is useful in analyzing phase shifts and describing resonances with accuracy in non-spherical scatterings.