Klein--Gordon oscillator with linear--fractional deformed Casimirs 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 $p^μ$ 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 $E\leftrightarrow -E$ 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 $\mathcal{PT}$-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.

Figures (2)

• Figure 1: Both energy branches $e_{n,\pm}=E_{n,\pm}/m$ versus $n$ for SR/spacelike, timelike/lightlike (first-power Casimir), and MS DSR (squared denominator), using \ref{['eq:plotparams']}. In SR/spacelike the spectrum is symmetric under $e\to -e$. In timelike/lightlike and MS the linear-in-$E$ deformation shifts both branches by the same additive amount, thereby breaking the exact $e_{n,-}=-e_{n,+}$ symmetry. The MS displacement is stronger at fixed $\epsilon$, consistent with the squared denominator.
• Figure 2: Energy shifts of the positive branch relative to SR, $\Delta e_{n,+}=e_{n,+}-e_{n,+}^{\mathrm{SR}}$, for the parameters \ref{['eq:plotparams']}. The spacelike geometry is exactly isospectral ($\Delta e_{n,+}=0$). Timelike and lightlike coincide and show a negative Planck-suppressed shift $\approx -\epsilon/2$. The MS model exhibits a larger negative shift, consistent with the squared-denominator Casimir and $\Delta e_{n,+}\simeq-\epsilon$ at leading order.