Three-dimensional modified dirac oscillator in standard and generalized doubly special relativity

Doubly Special Relativity (DSR) introduces, besides the invariant speed of light c , an observer-independent high-energy scale that deforms relativistic kinematics and can be implemented through modified dispersion relations or effective wave equations with energy-dependent spatial operators. In this work we develop a three-dimensional, exactly solvable benchmark for such deformations in the spin- 1/2 1 2 sector: the Dirac oscillator. Following the original construction of Moshinsky and Szczepaniak, the oscillator is introduced through a linear non-minimal momentum coupling, which preserves Hermiticity and yields, after decoupling the Dirac equation into large and small components, a three-dimensional isotropic harmonic-oscillator operator supplemented by a strong spin–orbit term. We then incorporate Planck-scale deformations in two standard DSR realizations (Amelino–Camelia and Magueijo–Smolin, characterized by an invariant energy scale k ) and in a generalized DSR framework based on a first-order expansion in the Planck length lₚ l p . In all cases the bound-state eigenfunctions retain the oscillator-spinor structure dictated by spherical symmetry, while DSR deforms the algebraic relation between quantum numbers (N,j, ) ( N , j , ℓ ) and the relativistic energy, producing branch-dependent shifts for both particle and antiparticle solutions. The undeformed limit ( k→ ∞ k → ∞ or lₚ→ 0 l p → 0 ) is recovered smoothly and the deformation signal increases with excitation through the oscillator scale and spin–orbit splitting.