Exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$

TL;DR

This paper provides an exact analytical solution to the one-dimensional Dirac equation with a $1/\sqrt{x}$ potential, detailing bound states and energy spectra for various field configurations.

Contribution

It introduces the first exact solutions for the 1D Dirac equation with an inverse-square-root potential across different field types.

Findings

Exact bound-state energy equations derived

Hermite functions form the fundamental solutions

Accurate approximations for energy spectra obtained

Abstract

We present the exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.