Non-hermitian Dirac Hamiltonian in the presence of local Fermi velocity

We present a new approach to study a class of non-Hermitian (1+1)-dimensional Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well known Nikiforov-Uvarov method to solve such a system. We discuss applications and explore the solvability of both $\mathcal{P}\mathcal{T}$-symmetric and non-$\mathcal{P}\mathcal{T}$ symmetric classes of potentials. In the former case we obtain the solution of a harmonic oscillator in the presence of a linear vector potential while in the latter case we solve the shifted harmonic oscillator problem.