Dirac oscillator in the background of a minimal surface

This study focuses on investigating the relativistic quantum dynamics of a fermionic oscillator field in the background of a minimal surface, specifically a catenoid bridge. Our primary objective is to analyze the bound-state solutions of this system by solving the non-perturbative radial wave equation. To achieve this, we consider two distinct scenarios. First, we explore the case where the particle is located inside the catenoid bridge and search for solutions to the wave equation. Subsequently, we extend our analysis to scenarios where the particle is positioned at a significant distance from the origin, comparable to the size of the catenoid bridge. In this regime, we obtain solutions to the wave equation to understand the behavior of the fermionic oscillator field. By examining these solutions, we investigate the influence of both the oscillator coupling and the spacetime background on the system.