Exact solutions of Dirac and Schrodinger equations for a large class of power-law potentials at zero energy

Abstract

We obtain exact solutions of Dirac equation at zero kinetic energy for radial power-law relativistic potentials. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrodinger equation with two-term power-law potentials at zero energy. The latter is solved by point canonical transformation of the exactly solvable problem of the three dimensional oscillator. The wavefunction 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 non-zero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.