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

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 \(l_p E \ll 1\).

Figures (6)

• Figure 1: SR energy spectrum in an infinite square well for both FV branches $E^{(\mathrm{SR})}_{n,\pm}$ as a function of $n$, for $(m=1,\;L=1)$ and $\ell_p\in\{0,\;0.02,\;0.2\}$. The curves overlap because $\ell_p$ does not enter the SR dispersion relation.
• Figure 2: DSR energy spectrum in an infinite square well for both FV branches $E^{(\mathrm{DSR})}_{n,\pm}$ as a function of $n$, for $(m=1,\;L=1)$ and $\ell_p\in\{0,\;0.02,\;0.2\}$. Increasing $\ell_p$ suppresses the magnitude of the levels and induces a high-$n$ saturation.
• Figure 3: G-DSR energy spectrum in an infinite square well for both FV branches $E^{(\mathrm{GDSR})}_{n,\pm}$ as a function of $n$, for $(m=1,\;L=1,\;\chi=1)$ and $\ell_p\in\{0,\;0.02,\;0.2\}$. The deformation lowers the energies relative to SR while preserving an unbounded spectrum.
• Figure 4: Standard (SR) case: transmission $T(E)$ (solid) and reflection $R(E)$ (dashed) in the FV formalism for $l_p\in\{0,0.02,0.06\}$. (For SR, the curves coincide because there is no deformation.) Parameters are given in \ref{['eq:numparams_Escan_FV']}.
• Figure 5: Standard DSR case: transmission $T(E)$ (solid) and reflection $R(E)$ (dashed) for $l_p\in\{0,0.02,0.06\}$, using the FV flux definition \ref{['eq:RT_def_FV']} and parameters \ref{['eq:numparams_Escan_FV']}.