ANALYSIS OF INVERSE-SQUARE POTENTIALS USING SUPERSYMMETRIC QUANTUM-MECHANICS

We analyse transition potentials, i.e. potentials exhibiting limiting inverse-square behaviour V(r)r approximately 0 --> alphar-2 in non-rel ativistic quantum mechanics using the techniques of supersymmetry. For the range - 1/4 less-than-or-equal-to alpha < 3/4, the eigenvalue pro blem 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. Inter estingly, our SUSYQM-based approach picks out the same 'less singular' wavefunctions as the conventional approach and thus provides a simple justification for the usual practice in the literature. Two examples (the Poschl-Teller II potential and a two-anyon system on the plane) h ave been worked out for illustrative purposes.