Non-Hermitian $\mathcal{PT}$ symmetric Klein-Gordon fields in Lorentz-violating wormholes spacetime background

Abstract

We investigate a $\mathcal{PT}$-symmetric Klein--Gordon (KG) oscillator in a $(3+1)$-dimensional Lorentz-violating (LV) traversable wormhole spacetime. The geometry is characterized by a smooth throat of radius $a$ and a constant lapse function, $\mathcal{A}(x)=1$, so that no Killing horizons occur and the two asymptotic regions at \(x\to \pm\infty\) remain connected through the throat. The Lorentz violation modifies the radial geometry and the oscillator scale according to $\widetilde{\Omega}=\Omega/\sqrt{1-\zeta}$. A nonminimal non-Hermitian coupling, $\mathcal{F}_t(x)=i\widetilde{\Omega}x$, leads to a $\mathcal{PT}$-symmetric KG oscillator with a regular effective potential at the throat and a quadratic confinement at large $|x|$. The radial equation can be mapped to the confluent Heun equation, and polynomial truncation gives a conditionally exactly solvable sector in which the oscillator frequency, throat radius, LV parameter, angular momentum, and radial quantum number satisfy algebraic constraints. For the degree-1 sector, these conditions restrict the allowed angular momenta and correlate them with the LV parameter. The resulting bound-state energies are real and discrete, with exact particle--antiparticle symmetry, $E^{-}=-E^{+}$, while the corresponding wave functions are regular and localized on the two-sided wormhole geometry. These results show how Lorentz violation and the $\mathcal{PT}$-symmetric coupling modify the relativistic bound-state spectrum through the geometry of the wormhole.