Klein-Gordon oscillators in traversable wormhole rainbow gravity spacetime: Conditional exact solvability via a throat radius and oscillator frequency correlation

In this study, we discuss an analytical solution for a set of the Klein-Gordon (KG) oscillators’ energies through a correlation between the frequency of the KG-oscillators and the traversable wormhole (TWH) throat radius. Under such restricted parametric correlation (hence the notion of conditionally exact solvability is unavoidable in the process), we report the effects of throat radius, rainbow parameter, disclination parameter, and oscillator frequency on the spectroscopic structure of a vast number of $(n,m)$-states (the radial and magnetic quantum numbers, respectively). In the process, we only use two loop quantum gravity motivated rainbow functions pairs. The only rainbow functions that clearly and reliably fully adhere to the rainbow gravity model and secure Planck energy $E_p$ as the maximum possible energy for particles and anti-particles alike. Near the asymptotically flat upper and lower universes connected by the TWH, i.e., for the throat radius $r_0>>1$, the energies tend to cluster around the rest mass energies, i.e., $\mathrm{\mid }{E}_{\pm }\mathrm{\mid }\sim m_0$. Whereas, for $r_0<<1$ the energies tend to approach $\mathrm{\mid }{E}_{\pm }\mathrm{\mid }\le E_P$.