Klein-Gordon Oscillator with Linear-Fractional Deformed Mass-Shell Invariants in Doubly Special Relativity

Abstract

We study the Klein--Gordon (KG) oscillator in a doubly special relativity (DSR) framework, where the mass-shell condition is deformed through a linear--fractional (Möbius-type) modification of the Casimir invariant. This is induced by a nonlinear map from physical momenta $p^{\mu }$ to auxiliary Lorentz-covariant variables ${\pi }^{\mu }$. In $(1+1)$ dimensions, the deformation is controlled by a constant covector $a_{\mu }$, yielding inequivalent realizations depending on whether $a_{\mu }$ is timelike, spacelike, or lightlike. Implementing the KG oscillator via a reverted-product nonminimal coupling, we obtain exact closed-form spectra and explicit eigensolutions for both particle and antiparticle branches across all three geometries. Timelike and lightlike deformations produce identical spectra characterized by a Planck-suppressed additive displacement. This breaks the exact $E\leftrightarrow -E$ symmetry via a term linear in $E$, interpretable as a branch-independent reparametrization of the energy origin. Conversely, the spacelike deformation is strictly isospectral to the undeformed oscillator but generates complex-shifted wavefunctions and a non-Hermitian spatial operator. We provide a compact $\mathcal{P}\mathcal{T}$-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map $\mathcal{S}$ to a Hermitian oscillator, deriving the metric operator $\eta ={\mathcal{S}}^{†}\mathcal{S}$, and establishing biorthonormal relations. Finally, we compare quantitatively with the Magueijo--Smolin (DSR2) model: the squared-denominator invariant leads to a larger Planck-suppressed displacement at fixed $m/E_{Pl}$, highlighting the denominator power's role in controlling spectral shifts. Representative plots illustrate the dependence on deformation ratio, oscillator strength, and excitation level.

Accepted for publication in International Journal of theoretical physics