# Position-dependent mass charged particles in magnetic and Aharonov-Bohm flux fields: separability, exact and conditionally exact solvability

## TL;DR

This paper investigates the quantum behavior of position-dependent mass charged particles in magnetic and flux fields, achieving exact and conditionally exact solutions using various potential models and differential equations.

## Contribution

It introduces new methods for solving the Schrödinger equation for PDM particles in complex fields, including exact and conditionally exact solutions with specific potential models.

## Key findings

- Achieved separability of the Schrödinger equation in cylindrical coordinates.
- Obtained exact solutions for PDM particles in magnetic and flux fields with certain potentials.
- Identified spectral signatures influenced by z-dependent potentials.

## Abstract

Using cylindrical coordinates, we consider position-dependent mass (PDM) charged particles moving under the influence of magnetic, Aharonov-Bohm flux, and a pseudoharmonic or a generalized Killingbeck-type potential fields. We implement the PDM-minimal-coupling recipe <cite>26</cite>, along with the PDM-momentum operator <cite>27</cite>, and report separability under radial cylindrical and azimuthal symmetrization settings. For the radial Schr\"odinger part, we transform it into a radial one-dimensional Schr\"odinger-type and use two PDM settings to report on the exact solvability of PDM charged particles moving in three fields: magnetic, Aharonov-Bohm flux, and pseudoharmonic potential fields. Next, we consider the radial Schr\"odinger part as is and use the biconfluent Heun differential forms for two PDM settings to report on the conditionally exact solvability of our PDM charged particles moving in three fields: magnetic, Aharonov-Bohm flux, and generalized Killingbeck potential fields. Yet, we report the spectral signatures of the one-dimensional z-dependent Schr\"odinger part on the overall eigenvalues and eigenfunctions, for all examples, using two z-dependent potential models (infinite potential well and Morse-type potentials).