Three-Dimensional Modified Klein--Gordon Oscillator in Standard and Generalized Doubly Special Relativity

Abstract: Doubly Special Relativity (DSR) augments special relativity by introducing, alongside the invariant speed of light $c$, a second observer-independent scale typically associated with the Planck regime. At the level of effective wave equations this principle manifests itself through deformed dispersion relations and energy-dependent spatial operators. Here we quantify such effects in a prototypical exactly solvable bound-state problem: the three-dimensional Klein--Gordon oscillator generated by a non-minimal momentum coupling that yields isotropic harmonic confinement while preserving rotational symmetry. We analyze two standard DSR realizations (Amelino--Camelia and Magueijo--Smolin, parametrized by an invariant energy scale $k$) as well as a generalized DSR framework based on a first-order expansion in the Planck length $l_p$. After stationary reduction and separation in spherical coordinates, the eigenfunctions retain the generalized-Laguerre and spherical-harmonic structure of the undeformed oscillator, whereas DSR deforms the algebraic quantization condition that relates the principal oscillator number $N=2n+\ell\in\mathbb{N}_0$ to the relativistic energy. Closed-form spectra are obtained for the standard DSR cases, and perturbative Planck-suppressed shifts are derived for the generalized model. In all realizations the deformation induces branch-dependent shifts of both positive- and negative-energy solutions, which increase with excitation and vanish smoothly in the limits $k\to\infty$ or $l_p\to0$. The main goal of this paper is to extract analytic spectra and Planck-suppressed shifts that enable a direct comparison between different DSR prescriptions in a fully three-dimensional setting.