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

TL;DR

This paper derives exact zero-energy solutions for the Dirac equation with power-law potentials, extending known Schrödinger solutions, and explores their properties including bound states and degeneracies.

Contribution

It provides a new class of exact solutions for the relativistic Dirac equation at zero energy for power-law potentials, linking them to Schrödinger solutions via transformations.

Findings

Solutions are expressed with confluent hypergeometric functions.

Most solutions support zero-energy bound states.

Degeneracy of nonrelativistic states is demonstrated.

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.