Extensions of the auxiliary field method to solve Schr\"{o}dinger equations

TL;DR

This paper extends the auxiliary field method to derive analytical solutions for Schrödinger equations with complex potentials, proving key properties and connecting it with perturbation theory.

Contribution

It introduces new extensions of the auxiliary field method for better analytical approximations of Schrödinger equations with combined potentials.

Findings

Proves scaling laws and independence of the auxiliary field method.

Derives analytical energy formulas for specific radial potentials.

Establishes connections between perturbation theory and the auxiliary field method.

Abstract

It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schr\"{o}dinger equation. This technique can generate the spectrum associated with an arbitrary potential $V(r)$ starting from the analytically known spectrum of a particular potential $P(r)$. In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of $P(r)$. The method is extended in order to find accurate analytical energy formulae for radial potentials of the form $a P(r)+V(r)$, and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.