Supersymmetric approach to exact solutions of $(1+1)$-dimensional time-independent Klein-Gordon equation : Application to a position-dependent mass and a $\mathcal{PT}$-symmetric vector potential

TL;DR

This paper applies supersymmetric quantum mechanics to find exact bound state solutions of the Klein-Gordon equation with position-dependent mass and $ ext{PT}$-symmetric potentials, providing conditions for physical solutions and exploring special cases.

Contribution

It introduces a SUSYQM-based method for solving the Klein-Gordon equation with complex potentials and position-dependent mass, extending analytical solutions in relativistic quantum mechanics.

Findings

Exact bound state solutions for models with position-dependent mass.

Conditions for physical solutions based on parameter constraints.

Derivation of special cases by parameter adjustment.

Abstract

Rigorous use of SUSYQM approach applied for Klein-Gordon equation with scalar and vector potentials is discussed. The method is applied to solve exactly, for bound states, two models with position-dependent masses and $\mathcal{PT}$-symmetric vector potentials, depending on some parameters. The necessary conditions on the parameters to get physical solutions are described. Some special cases are also derived by adjusting the parameters of the models.