Semirelativistic Hamiltonians and the auxiliary field method

TL;DR

This paper develops approximate analytical energy formulas for semirelativistic Hamiltonians using the auxiliary field method, applicable to various potentials, and compares these with exact solutions to validate the approach.

Contribution

It introduces a novel application of the auxiliary field method to semirelativistic Hamiltonians, providing analytical formulas for different potential shapes.

Findings

Derived approximate energy formulas for multiple potentials.

Validated formulas through detailed comparison with exact results.

Demonstrated the method's effectiveness for semirelativistic systems.

Abstract

Approximate analytical closed energy formulas for semirelativistic Hamiltonians of the form $\sigma\sqrt{\bm p^{2}+m^2}+V(r)$ are obtained within the framework of the auxiliary field method. This method, which is equivalent to the envelope theory, has been recently proposed as a powerful tool to get approximate analytical solutions of the Schr\"odinger equation. Various shapes for the potential $V(r)$ are investigated: power-law, funnel, square root, and Yukawa. A comparison with the exact results is discussed in detail.