Feshbach-Villars Hamiltonian Approach to the Klein-Gordon Oscillator and Supercritical Step Scattering in Standard and Generalized Doubly Special Relativity

• URL: http://arxiv.org/abs/2601.18836v1
• Date: Sun, 25 Jan 2026 20:57:45 GMT
• Title: Feshbach-Villars Hamiltonian Approach to the Klein-Gordon Oscillator and Supercritical Step Scattering in Standard and Generalized Doubly Special Relativity
• Authors: A. Boumali, N. Jafari, Y. Chargui,
• Abstract summary: We develop a first-order Feshbach-Villars framework for spin-0 relativistic quantum dynamics.<n>We study two benchmark settings in which the FV metric structure is essential.<n>For step and barrier scattering, we compute reflection and transmission coefficients directly from the pseudo-Hermitian FV current.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We develop a first-order Feshbach-Villars (FV) Hamiltonian framework for spin-0 relativistic quantum dynamics in the presence of Planck-scale kinematic deformations described within generalized doubly special relativity (G-DSR). Starting from a generic nonlinear momentum-space map, we derive the corresponding modified dispersion relation (MDR) at leading order in the Planck length \(l_p\) and construct a consistent FV linearization of the deformed Klein-Gordon operator. The resulting two-component Hamiltonian remains \(σ_3\)-pseudo-Hermitian at \(\mathcal{O}(l_p)\), which guarantees conservation of the FV charge and current and provides a current-based definition of reflection and transmission in stationary scattering. As applications, we study two benchmark settings in which the FV metric structure is essential: (i) the one-dimensional Klein-Gordon oscillator and (ii) scattering from electrostatic step and barrier potentials. For the oscillator, we obtain controlled \(\mathcal{O}(l_p)\) branch-resolved spectral shifts and show how kinetic versus mass-shell deformations reshape the level spacing and the high-energy spectral compression. For step and barrier scattering, we compute reflection and transmission coefficients directly from the pseudo-Hermitian FV current and quantify the deformation-induced shift of the supercritical (pair-production) threshold. A comparative analysis of the Amelino-Camelia and Magueijo-Smolin realizations indicates that MS-type deformations generally delay the onset of the supercritical regime and reduce the magnitude of the negative transmitted flux within the validity domain \(l_p E \ll 1\).

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