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

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.