Real exact solutions to the relativistic spinless particle with modified Rosen-Morse potential and its Pscr Tscr symmetry

The real exact solutions to the relativistic spinless particle with modified Rosen-Morse potential are presented. The novel issue is that the previously obtained eigenfunctions related to this type of potential in an imaginary variable i sinh (α x) (Dirac case) can be expressed by the real Nomanovski polynomials. The energy levels are calculated numerically. It is interesting to note that the energy levels first increase with the parameter V ₁ (V ₁<0) and then decrease with it. In the case of the Pscr Tscr symmetric version it is found that the eigenvalues are real even though the Pscr Tscr symmetric potentials are non-Hermitian.