Effects of position-dependent mass (PDM) on the bound-state solutions of a massive spin-0 particle subjected to the Yukawa potential

TL;DR

This paper investigates how position-dependent mass influences the bound-state solutions of a massive spin-0 particle in a Yukawa potential, revealing effects like energy gap closure and spectrum modifications.

Contribution

It introduces the analysis of PDM effects on Klein-Gordon bound states with Yukawa potential, highlighting phenomena not observed with constant mass.

Findings

PDM causes equivalence of positive and negative energy states at low energies.

PDM can induce gap closure at a critical Yukawa potential strength.

Energies can be forced to zero at zero shielding factor due to PDM.

Abstract

With the advent of Albert Einstein's theory of special relativity, Klein and Gordon made the first attempt to elevate time to the status of a coordinate in the Schr\"odinger equation. In this study, we graphically discuss the eigenfunctions and eigenenergies of the Klein-Gordon equation with a Yukawa-type potential (YP), within a position-dependent mass (PDM) framework. We conclude that the PDM leads to the equivalence of the positive ($E^+$) and negative ($E^-$) solution states at low energies. We observe that in the energy spectrum as a function of $\eta$ (YP intensity factor), the PDM can induce gap closure at the critical point where $E^+$ and $E^-$ become imaginary. In the spectrum as a function of $\alpha$ (YP shielding factor), it can compel the energies to be zero at $\alpha=0$, instead of being equal to $(m_0c^2)$ as in the invariant mass case.