Klein--Gordon oscillator with linear--fractional deformed Casimirs in doubly special relativity

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^μ to auxiliary Lorentz-covariant variables π^μ. In (1+1) dimensions, the deformation is controlled by a constant covector a_μ, yielding inequivalent realizations depending on whether a_μ 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 Eleftrightarrow -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{PT}-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map mathcal{S} to a Hermitian oscillator, deriving the metric operator η=mathcal{S}^dagger 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.