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).
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
pμ to auxiliary Lorentz-covariant variables
πμ. In
(1+1) dimensions, the deformation is controlled by a constant covector
aμ, yielding inequivalent realizations depending on whether
aμ 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↔−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
PT-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map
S to a Hermitian oscillator, deriving the metric operator
η=S†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/EPl, highlighting the denominator power's role in controlling spectral shifts. Representative plots illustrate the dependence on deformation ratio, oscillator strength, and excitation level.
We present a unified analytical and numerical study of the one-dimensional Feshbach--Villars (FV) equation for spin-0 particles in the presence of several representative external potentials. Starting from the FV formulation of the Klein--Gordon equation, we derive the corresponding one-dimensional master equation and analyse its solutions for Coulomb, power-exponential, Cornell, Pöschl--Teller, and Woods--Saxon interactions. For the singular Coulomb and Cornell cases, a Loudon-type cutoff regularisation is implemented on the full line, allowing a mathematically controlled treatment of the origin and an explicit classification of the states by parity. The Coulomb problem exhibits the expected near-degenerate even--odd structure in the cutoff limit, while the Cornell potential combines short-distance Coulomb behaviour with long-distance confinement and produces a finite set of bound states for fixed parameters. The power-exponential potential with
p=1 is reduced to a Whittaker-type equation and yields an intrinsically relativistic spectrum with no standard Schrödinger bound-state limit in the parameter regime considered. For the smooth short-range Pöschl--Teller and Woods-Saxon potentials, the FV formalism reveals, respectively, the effects of definite parity and spatial asymmetry on the spectrum, wave functions, and particle--antiparticle mixing. In all cases, we reconstruct the full FV spinor, analyse the associated charge density, and compare the relativistic behaviour with the corresponding non-relativistic expectations whenever such a limit exists. The results provide a coherent set of analytical and numerical benchmarks for relativistic scalar bound states in one dimension.
In this study, a relativistic formulation of the
(q)-deformed Dunkl-Fokker-Planck equation in
(1+1)-dimensions is constructed within the reflection-deformed quantum framework. In this case, the formalism includes
(q)-deformed Dunkl operators and reflection symmetry to build a generalized dynamical structure for a relativistic quantum systems framework. Moreover, the corresponding
(q)-Wigner-Dunkl supersymmetric configuration is established via the construction of deformed ladder operators and supersymmetric algebraic relations, yielding a consistent spectral representation of the model within the algebraic framework. The analysis extends to the harmonic oscillator with centrifugal interaction, where exact algebraic solutions, similarity reduction techniques, and closed energy spectra are obtained analytically in detail. The role of the deformation parameter and reflection operator on spectral properties and wavefunction structure is examined systematically in detail. A generalized Foldy-Wouthuysen (FW) transformation is introduced within the deformed Dunkl framework to achieve relativistic decoupling of positive- and negative-energy sectors within the present theoretical formulation. In this case, this approach yields an effective reduced Hamiltonian, including higher-order relativistic and deformation-induced terms. Also, the associated Dunkl-Fokker-Planck dynamics generated through high-order FW reduction are examined in detail for reflection-deformed relativistic quantum systems. In this context, results obtained here yield a unified algebraic and relativistic description of
(q)-deformed Dunkl structures and construct a consistent framework for investigating supersymmetric and relativistic properties in reflection-symmetric quantum models in general.
We consider the relativistic quantum scattering of spinless particles in one-spatial dimension using the Feshbach-Villars formalism. We construct the general form of the scattering matrix, for symmetric and non-symmetric potentials, based on the symmetry properties of the Feshbach-Villars equation. Then, since in one dimension there are only two partial waves associated with even and odd parities, we show, in a simple and comprehensive way, how to describe this transmission-reflection problem using partial-wave decomposition. As an illustration, we also discuss the special case of scattering by a symmetric square-well potential.
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p2∕2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator η satisfying H†=ηHη−1 to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate η for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general η up to second-order terms in the coupling constants.
The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.