Non-Hermitian 𝒫𝒯 symmetric Klein-Gordon fields in Lorentz-violating wormholes spacetime background

We investigate a [Formula: see text]-symmetric Klein–Gordon (KG) oscillator in a [Formula: see text]-dimensional Lorentz-violating (LV) traversable wormhole spacetime. The geometry is characterized by a smooth throat of radius [Formula: see text] and a constant lapse function, [Formula: see text], so that no Killing horizons occur and the two asymptotic regions at [Formula: see text] remain connected through the throat. The Lorentz violation modifies the radial geometry and the oscillator scale according to [Formula: see text]. A nonminimal non-Hermitian coupling, [Formula: see text], leads to a [Formula: see text]-symmetric KG oscillator with a regular effective potential at the throat and a quadratic confinement at large [Formula: see text]. 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 exact and conditionally-exact bound-state energies are real and discrete, with exact particle–antiparticle symmetry, [Formula: see text], while the corresponding wave functions are regular and localized on the two-sided wormhole geometry. An independent finite-difference diagonalization of the radial operator confirms the degree-1 analytical eigenvalue with the expected second-order convergence and high numerical accuracy. These results show how Lorentz violation and the [Formula: see text]-symmetric coupling modify the relativistic bound-state spectrum through the geometry of the wormhole.