An Alternate Approach to Transition Potentials

Abstract

We analyze transition potentials $(V(r) \stackrel{r\sim 0}{\rightarrow} {\alpha r^{-2}})$ in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range $-1/4 < \alpha < 3/4$, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the P\"oschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes.