Exact solutions of the Dirac Hamiltonian on the sphere under hyperbolic magnetic fields

TL;DR

This paper derives exact solutions for the Dirac Hamiltonian on a spherical surface influenced by hyperbolic magnetic fields, utilizing supersymmetry and special potential models to find energy spectra and eigenstates.

Contribution

It introduces a method to obtain exact solutions of the Dirac equation on a sphere with hyperbolic magnetic fields using supersymmetric partner Hamiltonians and specific solvable potentials.

Findings

Exact energy spectrum for Rosen Morse II potential

Complete solutions for the second potential model

Eigenstates and potentials explicitly derived

Abstract

Two dimensional massless Dirac Hamiltonian under the influence of hyperbolic magnetic fields is mentioned in curved space. Using a spherical surface parametrization, the Dirac operator on the sphere is presented and the system is given as two supersymmetric partner Hamiltonians which coincides with the position dependent mass Hamiltonians. We introduce two ansatzes for the component of the vector potential to acquire effective solvable models, which are Rosen Morse II potential and the model given in \cite{mid} whose bound states are Jacobi $X_1$ type polynomials, and we adapt our work to these special models under some parameter restrictions. The energy spectrum and the eigenvectors are found for Rosen Morse II potential. On the other hand, complete solutions are given for the second system. The vector and the effective potentials with their eigenvalues are sketched for each system.