Relativistic quantum effects in compact astrophysical objects: a harmonic oscillator approach

In this study, we examine relativistic quantum harmonic oscillators and their applications to compact astrophysical objects. Using a refined Hermite polynomial series approach, we derive analytical solutions for energy eigenvalues and normalized eigenfunctions to order $v^2/c^2$, extending beyond previous approximations. A computational model, validated against these analytical results, explores neutron star and black hole parameters. Several percent of the total energies of these compact objects are affected by relativistic corrections. Our model predicts relativistic corrections of up to 20\% for oscillation frequencies in highly magnetized neutron stars and modifications of 1-5\% to innermost stable circular orbits around stellar-mass black holes. As compared to non-relativistic equations of state, we observe measurable deviations in mass-radius relations in neutron stars. These modifications have implications for accretion disk dynamics and associated emissions, potentially detectable with next-generation X-ray timing observations.