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.

Figures (2)

• Figure 1: Energy spectrum $E_N$ as a function of the principal oscillator number $N$ for the Klein--Gordon oscillator, showing both branches: the positive-energy branch $E_N^{(+)}$ (solid lines) and the negative-energy branch $E_N^{(-)}$ (dashed lines). The undeformed result $E_N^{(0)}=\pm\sqrt{m^2c^4+2mc^2\hbar\omega\,N}$ is compared with representative DSR-deformed realizations (standard AC-DSR, standard MS-DSR, and generalized DSR in AC/MS-type first-order $l_p$ expansions). Here $N\in\mathbb{N}_0$ is an integer, $N=0,1,2,\dots$.
• Figure 2: Relative spectral shift $\delta_N\equiv(E_N-E_N^{(0)})/E_N^{(0)}$ as a function of $N$ for both branches. Solid (dashed) curves correspond to the positive- (negative-) energy branch, and the denominator uses the matching undeformed branch $E_N^{(0)}$ so that $E_N^{(0)}<0$ for the negative branch. The figure isolates the deformation signal and highlights its growth with excitation number $N\in\mathbb{N}_0$ (integer $N=0,1,2,\dots$).