Spherical-separability of Non-Hermitian Hamiltonians and Pseudo-PT -symmetry

Abstract

Non-Hermitian but PϕTϕ-symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians Hr , Hθ , and Hϕ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a PϕTϕ-symmetrized Hϕ, we have shown that the conventional Dirac (relativistic) and Schrödinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V (θ) = 0 in the descendant Hamiltonian Hθ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some PϕTϕ-symmetrized Hϕ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the PT -symmetric ones (here the non-Hermitian PϕTϕ-symmetric Hamiltonians) are nicknamed as pseudo- PT -symmetric.