$\mathcal{PT}$ symmetric Klein-Gordon oscillators in Lorentz-violating wormholes

Abstract

We study spin-0 $\mathcal{PT}$-symmetric Klein-Gordon (KG) oscillator fields in a (3+1)-dimensional Lorentz-violating (LV) traversable wormhole background. The wormhole geometry, characterized by a smooth minimal throat $a$ and a regular lapse sector, induces a curvature-driven deformation of the relativistic quantum dynamics under Lorentz symmetry breaking. A nonminimally coupled non-Hermitian \(\mathcal{PT}\) symmetric scalar bosonic field $\mathcal{F}_t(x)=i\tildeΩx$, with $\tildeΩ=Ω/\sqrt{1-ζ}$, generates a globally regular effective \(\mathcal{PT}\) symmetric KG oscillator free of centrifugal singularities. The radial equation reduces to a confluent Heun system, admitting bound states only under polynomial truncation conditions that impose constraints among $(Ω,a,ζ,n,\ell)$. The resulting energy spectrum is real, discrete and exactly symmetric under $E \rightarrow -E$, determined by spacetime curvature, Lorentz-violation parameter $ζ$, and oscillator strength $Ω$, within a conditionally exactly solvable framework.