Auxiliary field method and analytical solutions of the Schr\"{o}dinger equation with exponential potentials

TL;DR

This paper extends the auxiliary field method to analytically solve the Schrödinger equation with exponential potentials, providing formulas for energy levels and critical heights, including for Yukawa and pure exponential potentials.

Contribution

It demonstrates the application of the auxiliary field method to exponential potentials, deriving analytical solutions and formulas for energy spectra.

Findings

Analytical solutions for exponential potentials are obtained.

Formulas for critical heights and energy levels are provided.

Special cases like Yukawa and pure exponential potentials are analyzed.

Abstract

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schr\"{o}dinger equation with exponential potentials of the form $-\alpha r^\lambda \exp(-\beta r)$ can also be analytically solved by using the auxiliary field method. Formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn on the Yukawa potential and the pure exponential one.