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 $p^{\mu }$ 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 $m\mathrm{/}{E}_{Pl}$, highlighting the denominator power’s role in controlling spectral shifts. Representative plots illustrate the dependence on deformation ratio, oscillator strength, and excitation level.