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

We investigate non-Hermitian $\mathcal{PT}$-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 $\mathcal{PT}$-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 $\mathcal{PT}$-symmetry in regulating the allowed spectrum.