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 $\mathcal{PT}$-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map $\mathcal{S}$ to a Hermitian oscillator, deriving the metric operator $\eta=\mathcal{S}^\dagger \mathcal{S}$, 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.

arXiv:2603.06637 [quant-ph]

(or arXiv:2603.06637v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2603.06637 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.1007/s10773-026-06464-5 Focus to learn more DOI(s) linking to related resources

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