Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.06717

Nonlocal Generalized Dirac Oscillators in (1 + 1) Dimensions

Abdelmalek Boumali

quant-phhep-thSubmitted Mar 5, 2026• Announced Mar 9, 2026• 0 citations
arXivHTMLPDF

Abstract

We propose a nonlocal extension of the generalized Dirac oscillator (GDO) in $(1+1)$ dimensions by replacing the multiplicative interaction $f(x)$ with an integral operator $\hat F$ with kernel $f(x,x')$. The resulting Dirac equation preserves an operator factorization and decouples into two nonlocal Schrödinger-type (Sturm--Liouville) equations for the spinor components. We derive explicit expressions for the associated supersymmetric partner kernels in terms of $f$ and its derivatives, and we show that a complex-translation metric $η=e^{-θp_x}$ leads to a simple sufficient \emph{kernel-level} pseudo-Hermiticity constraint, $f(x+\ii\hbarθ,x'+\ii\hbarθ)=f^*(x',x)$, extending the familiar local complex-shift criteria. To provide a transparent \emph{nonlocal-to-local} interpretation, we adapt the Coz--Arnold--MacKellar current-based localization to each component equation, obtaining energy-dependent equivalent local potentials and multiplicative Perey (damping) factors. The mapping breaks down precisely at current zeros, thereby diagnosing the spurious solutions of the corresponding nonlocal Schrödinger problem. Finally, we illustrate the formalism with analytically tractable benchmarks (the local Dirac oscillator and a translation-invariant kernel) and with a finite-rank separable model (Gaussian form factor) that reduces the integro-differential problem to a small set of coupled ordinary differential equations and algebraic constraints.

Related papers

Algebraic approach for the one-dimensional Dirac–Dunkl oscillator

D. Ojeda-Guillén, R. D. Mota, M. Salazar–Ramírez +1 more

We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.

arXiv:2003.08975•Modern Physics Letters A•Aug 7, 2020•43 citations

A Non-Hermitian Relativistic Oscillator Exactly Mapped to a Hermitian Schrodinger Problem

Mustapha Maamache

We introduce the Dirac Inverted Oscillator, a new exactly solvable relativistic quantum system obtained through a Hermitian modification of the generalized momentum operator in the Dirac equation. In contrast to the conventional Dirac oscillator, where the generalized momentum is intrinsically non-Hermitian while the Hamiltonian remains Hermitian, the present construction reverses these Hermiticity properties: the generalized momentum becomes Hermitian, whereas the corresponding Dirac Hamiltonian is intrinsically non-Hermitian. This inversion leads to a new class of relativistic Hamiltonians within the framework of non-Hermitian quantum mechanics. Starting from the stationary Dirac equation, we derive the corresponding second-order wave equation and establish the generalized symmetry properties of the model, including pseudo-Hermiticity and pseudo- symmetry. We then prove that the Dirac Inverted Oscillator is exactly related to the conventional Dirac oscillator through a Hermitian but non-unitary similarity transformation generated by a dilation operator. This transformation is bijective and invertible, mapping the original non-Hermitian relativistic problem onto an equivalent Hermitian Schrodinger eigenvalue problem while preserving its complete spectral structure. The transformed Hermitian problem is solved analytically, allowing the exact relativistic spectrum and the corresponding eigenfunctions of the original non-Hermitian Hamiltonian to be determined explicitly. The present work therefore establishes an exact correspondence between a non-Hermitian relativistic Dirac system and a Hermitian Schrodinger problem, providing a new exactly solvable model and illustrating the effectiveness of Hermitian non-unitary similarity transformations for constructing analytically solvable relativistic Hamiltonians beyond the conventional Hermitian framework.

quant-pharXiv:2606.15303•Jun 16, 2026•0 citations

Supersymmetric Dirac Hamiltonians in ($$1+1$$) dimensions revisited

Georg Junker

Abstract The most general Dirac Hamiltonians in $$(1+1)$$ ( 1 + 1 ) dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example, the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed-form expression by utilising the energy-dependent Green’s function of the non-relativistic harmonic oscillator. The supersymmetric quasi-classical approximation for the Witten model is extended to the associated relativistic model.

arXiv:1911.00454•The European Physical Journal Plus•Jun 1, 2020•17 citations

A generalized harmonic oscillator problem for a spin-1/2 fermion

V. B. Mendrot, A. S. de Castro, P. Alberto

The exact bound-state wavefunctions and the corresponding energy equation are calculated for a new generalized harmonic oscillator problem describing a spin-1/2 fermion in 3+1-dimensions, involving scalar, vector, and tensor couplings acting simultaneously within a particular plane of motion. For the scalar and vector coupling, singular harmonic oscillator shapes are considered, such that the singular term is needed to allow analytical solutions for the wavefunctions while preserving binding under adequate conditions. For the tensor sector,the Dirac oscillator potential is employed, which adds another independent binding mechanism to the problem. The exact bound-state solutions are computed by specifically tuning coefficients for an appropriate pair of \textit{Ansätze} for the radial functions, which leads to wavefunctions in terms of generalized Laguerre polynomials. Although the energy equation cannot provide a general expression for the energy spectrum, specific constraints on the quantum numbers as simple functions of the external potential parameters can be derived for it, which determines the conditions for bound solutions to exist, and of what type: particle, antiparticle or both. It is shown that the results can be simply mapped to the spherically symmetric analogue problem, and this is used to show that the general result encompass several previous particular cases of spherically symmetric harmonic oscillator problems in the Dirac equation available in the literature.

quant-pharXiv:2609.02043•Sep 3, 2026•0 citations

Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics

Camila C. Soares, Luis B. Castro, Antonio S. de Castro

We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.

quant-pharXiv:2607.00306•Jul 2, 2026

Dirac oscillator in a helically twisted spacetime with axial torsion

Matheus D. Moro, Fabiano M. Andrade, Faizuddin Ahmed +1 more

We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix $β$ in the radial Moshinsky coupling, we show that the second-order problem is the ordered product $\hatΠ_+\hatΠ_-$ rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination $m/r-ωk$ and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at $E=Mc^2$. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.

hep-thCross-listedarXiv:2607.01301•Jul 2, 2026
We propose a nonlocal extension of the generalized Dirac oscillator (GDO) in (1+1)(1+1)(1+1) dimensions by replacing the multiplicative interaction f(x)f(x)f(x) with an integral operator F^\hat FF^ with kernel f(x,x′)f(x,x')f(x,x′). The resulting Dirac equation preserves an operator factorization and decouples into two nonlocal Schrödinger-type (Sturm--Liouville) equations for the spinor components. We derive explicit expressions for the associated supersymmetric partner kernels in terms of fff and its derivatives, and we show that a complex-translation metric η=e−θpxη=e^{-θp_x}η=e−θpx​ leads to a simple sufficient \emph{kernel-level} pseudo-Hermiticity constraint, f(x+\iiℏθ,x′+\iiℏθ)=f∗(x′,x)f(x+\ii\hbarθ,x'+\ii\hbarθ)=f^*(x',x)f(x+\iiℏθ,x′+\iiℏθ)=f∗(x′,x), extending the familiar local complex-shift criteria. To provide a transparent \emph{nonlocal-to-local} interpretation, we adapt the Coz--Arnold--MacKellar current-based localization to each component equation, obtaining energy-dependent equivalent local potentials and multiplicative Perey (damping) factors. The mapping breaks down precisely at current zeros, thereby diagnosing the spurious solutions of the corresponding nonlocal Schrödinger problem. Finally, we illustrate the formalism with analytically tractable benchmarks (the local Dirac oscillator and a translation-invariant kernel) and with a finite-rank separable model (Gaussian form factor) that reduces the integro-differential problem to a small set of coupled ordinary differential equations and algebraic constraints.
We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.
We introduce the Dirac Inverted Oscillator, a new exactly solvable relativistic quantum system obtained through a Hermitian modification of the generalized momentum operator in the Dirac equation. In contrast to the conventional Dirac oscillator, where the generalized momentum is intrinsically non-Hermitian while the Hamiltonian remains Hermitian, the present construction reverses these Hermiticity properties: the generalized momentum becomes Hermitian, whereas the corresponding Dirac Hamiltonian is intrinsically non-Hermitian. This inversion leads to a new class of relativistic Hamiltonians within the framework of non-Hermitian quantum mechanics. Starting from the stationary Dirac equation, we derive the corresponding second-order wave equation and establish the generalized symmetry properties of the model, including pseudo-Hermiticity and pseudo- symmetry. We then prove that the Dirac Inverted Oscillator is exactly related to the conventional Dirac oscillator through a Hermitian but non-unitary similarity transformation generated by a dilation operator. This transformation is bijective and invertible, mapping the original non-Hermitian relativistic problem onto an equivalent Hermitian Schrodinger eigenvalue problem while preserving its complete spectral structure. The transformed Hermitian problem is solved analytically, allowing the exact relativistic spectrum and the corresponding eigenfunctions of the original non-Hermitian Hamiltonian to be determined explicitly. The present work therefore establishes an exact correspondence between a non-Hermitian relativistic Dirac system and a Hermitian Schrodinger problem, providing a new exactly solvable model and illustrating the effectiveness of Hermitian non-unitary similarity transformations for constructing analytically solvable relativistic Hamiltonians beyond the conventional Hermitian framework.
Supersymmetric Dirac Hamiltonians in (1+11+11+1) dimensions revisited
Abstract The most general Dirac Hamiltonians in (1+1)(1+1)(1+1) ( 1 + 1 ) dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example, the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed-form expression by utilising the energy-dependent Green’s function of the non-relativistic harmonic oscillator. The supersymmetric quasi-classical approximation for the Witten model is extended to the associated relativistic model.
The exact bound-state wavefunctions and the corresponding energy equation are calculated for a new generalized harmonic oscillator problem describing a spin-1/2 fermion in 3+1-dimensions, involving scalar, vector, and tensor couplings acting simultaneously within a particular plane of motion. For the scalar and vector coupling, singular harmonic oscillator shapes are considered, such that the singular term is needed to allow analytical solutions for the wavefunctions while preserving binding under adequate conditions. For the tensor sector,the Dirac oscillator potential is employed, which adds another independent binding mechanism to the problem. The exact bound-state solutions are computed by specifically tuning coefficients for an appropriate pair of \textit{Ansätze} for the radial functions, which leads to wavefunctions in terms of generalized Laguerre polynomials. Although the energy equation cannot provide a general expression for the energy spectrum, specific constraints on the quantum numbers as simple functions of the external potential parameters can be derived for it, which determines the conditions for bound solutions to exist, and of what type: particle, antiparticle or both. It is shown that the results can be simply mapped to the spherically symmetric analogue problem, and this is used to show that the general result encompass several previous particular cases of spherically symmetric harmonic oscillator problems in the Dirac equation available in the literature.
We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.
We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix βββ in the radial Moshinsky coupling, we show that the second-order problem is the ordered product Π^+Π^−\hatΠ_+\hatΠ_-Π^+​Π^−​ rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination m/r−ωkm/r-ωkm/r−ωk and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at E=Mc2E=Mc^2E=Mc2. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.