Relativistic confinement of neutral fermions with partially exactly solvable and exactly solvable PT-Symmetric potentials in the presence of position-dependent mass

The relativistic problems of neutral fermions subject to a new partially exactly solvable PT-symmetric potential and an exactly solvable PT-symmetric hyperbolic cosecant potential in 1+1 dimensions are investigated. The Dirac equation with the double-well-like mass distribution in the background of the PT-symmetric vector potential coupling can be mapped into the Schrodinger-like equation with the partially exactly solvable double-well potential. The position-dependent effective mass Dirac equation with the PT-symmetric hyperbolic cosecant potential can be mapped into the Schrodinger-like equation with the exactly solvable modified Poschl-Teller potential. The real relativistic energy levels and corresponding spinor wavefunctions for the bound states have been given in a closed form.