Spectroscopic structure of Non-Hermitian $\mathcal{P}\mathcal{T}$-Symmetric Klein--Gordon Fields in a Magnetized Cosmic String Spacetime

Abstract

We investigate non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Klein--Gordon (KG) fields in a magnetized cosmic string spacetime. A complex non-minimal scalar interaction, ${\mathcal{F}}_{\mu }=(0,{\mathcal{F}}_r,0,0)$ with ${\mathcal{F}}_r=\omega r+ib\in \mathbb{C}$, indulged with a magnetic charge $q=ie$, is shown to introduce an effective $\mathcal{P}\mathcal{T}$-symmetric Klein--Gordon oscillator supplemented by complex Coulombic and linear interactions. The resulting radial equation is shown to be conditionally exactly solvable using a biconfluent Heun series-to-polynomial approach. To observe non-Hermitian \(\mathcal{PT}\) symmetrization, we compare with the Hermitian counterpart by mapping $e=-i\tilde{e}\Rightarrow q\in \mathbb{R}$ and $b=-i\tilde{b}\Rightarrow {\mathcal{F}}_r\in \mathbb{R}$. In the limiting case $\omega =0$, the system is shown to reduce to an exactly solvable non-Hermitian \(\mathcal{PT}\) symmetric Coulomb-type KG equation using confluent hypergeometric series/polynomials. Hereby, for both \(\mathcal{PT}\) symmetric models considered, we show that non-Hermitian \(\mathcal{PT}\) symmetrization introduces an upper limit for the allowed energies, unlike the corresponding Hermitian cases. These results provide an analytically tractable framework for exploring non-Hermitian relativistic quantum fields in curved spacetimes and demonstrate the role of $\mathcal{P}\mathcal{T}$-symmetry in regulating the allowed spectrum.

7 pages, 1 Figure