Mapping the charge-dyon system into the position-dependent effective mass background via Pauli equation

This work aims to reproduce a quantum system composed of a charged spin - $1/2$ fermion interacting with a dyon with an opposite electrical charge (charge-dyon system), utilizing a position-dependent effective mass (PDM) background in the non-relativistic regime via the PDM free Pauli equation. To investigate whether there is a PDM quantum system with the same physics (analogous model) that a charge-dyon system (target system), we resort to the PDM free Pauli equation itself. We proceed with replacing the exact bi-spinor of the target system into this equation, obtaining an uncoupled system of non-linear partial differential equations for the mass distribution $M$. We were able to solve them numerically for $M$ considering a radial dependence only, i.e., $M=M(r)$, fixing $\theta_0$, and considering specific values of $\mu$ and $m$ satisfying a certain condition. We present the solutions graphically, and from them, we determine the respective effective potentials, which actually represent our analogous models. We study the mapping for eigenvalues starting from the minimal value $j = \mu - 1/2$.