EXACT SOLUTIONS OF DIRAC AND SCHRÖDINGER EQUATIONS FOR A LARGE CLASS OF POWER-LAW POTENTIALS AT ZERO ENERGY

Abstract

We obtain exact solutions of Dirac equation for radial power-law relativistic potentials at rest mass energies. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrödinger equation at zero energy carrying representations of SO(2,1) Lie algebra. The latter are obtained by point canonical transformations of the exactly solvable problem of the three dimensional oscillator. The wave function solutions are written in terms of the confluent hypergeometric functions and almost always square integrable. For most cases these solutions support bound states at zero energy. Some exceptional unbounded states are normalizable for nonzero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.