Dirac and Klein – Gordon particles in complex Coulombic fields

The observation that the existence of the amazing reality and discreteness of the spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian is reemphasized in the context of the non-Hermitian Dirac and Klein-Gordon Hamiltonians. Complex Coulombic potentials are considered. In one of the first explicit studies of the non-Hermitian Schrödinger Hamiltonians, Caliceti et al [1] have considered the imaginary cubic oscillator problem in the context of perturbation theory. They have offered the first rigorous explanation why the spectrum in such a model may be real and discrete. Only many years later, after being quoted as just a mathematical curiosity [2] in the literature, the possible physical relevance of this result reemerged and emphasized [3]. Initiating thereafter an extensive discussion which resulted in the proposal of the so called PT −symmetric quantum mechanics by Bender and Boettcher [4]. The spiritual wisdom of the new formalism lies in the observation that the existence of the real spectrum need not necessarily be attributed to the Hermiticity of the Hamiltonian. This observation has offered a sufficiently strong motivation for the continued interest in the complex, non-Hermitian, cubic model which may be understood as a characteristic representation of a very broad class of the so-called pseudo-Hermitian models with real spectra. In such non-Hermitian settings, new intensive studies employed , for example, the idea of the strong coupling expansion [5], the complex WKB [6], Hill determinants and Fourier transformation [7], functional analysis [8], variational and truncation techniques [9], linear programing [10], pseudo-perturbation technique [11,12], ..etc (cf [13-15]). However such studies remain in the context of Schrödinger Hamiltonian and need to be complemented by the non-Hermitian setting of Dirac and Klein-Gordon Hamiltonians. Starting, say, with our forthcoming oversimplified generalized complex Coulombic examples. A priori, a generalized Dirac Coulomb equation for a mixed potential consists of a Lorentz-scalar Coulomb-like and a Lorentz-vector Coulomb potentials. Whilst the former is