PT symmetric Hamiltonian model and Dirac equation in 1+1 dimensions

In this paper, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as (H) over cap = omega((b) over cap dagger(b) over cap +1/2)+alpha((b) over cap (2)-((b) over cap dagger)(2)) where omega and alpha are real constants, (b) over cap and (b) over cap dagger are first-order differential operators. The Hermitian form of the Hamiltonian (H) over cap is obtained by suitable mappings and it is interrelated to the time-independent one-dimensional Dirac equation in the presence of position-dependent mass. Then, Dirac equation is reduced to a Schrodinger-like equation and two new complex non-PT symmetric vector potentials are generated. We have obtained a real spectrum for these new complex vector potentials using the shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.