Generalized Dirac Oscillator

• Generalized Dirac oscillator is a relativistic model that replaces standard linear coupling with a general interaction function, enabling analysis via supersymmetric techniques.
• It supports both Hermitian and pseudo-Hermitian frameworks, yielding exact spectral solutions for models such as Morse and isotonic oscillators.
• Extensions to curved-space, deformed kinematics, and entanglement studies demonstrate its versatility and impact in advanced quantum mechanics research.

Searching arXiv for recent and foundational papers on the generalized Dirac oscillator. The generalized Dirac oscillator is a relativistic oscillator model obtained by replacing the linear non-minimal coupling of the standard Dirac oscillator with a more general interaction function. In $(1+1)$ dimensions this is commonly written as the replacement $m\omega x\to W(x)$, leading to

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$

or equivalently, in the notation $f(x)$,

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$

with $c=1$ in the latter convention. In this sense, the standard Dirac oscillator is the special case $W(x)=m\omega x$ or $f(x)=m\omega x$, whereas the generalized model admits arbitrary local couplings, complex interactions, nonlocal kernels, curved-space versions, and deformed kinematics, while still often retaining an exactly solvable supersymmetric structure (Laba et al., 5 Jul 2026, Boumali, 3 Mar 2026, Dutta et al., 2013).

1. Defining equations and canonical formulations

The standard starting point is the Moshinsky–Szczepaniak substitution

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$

which preserves linearity in momentum and coordinate. The generalized Dirac oscillator replaces the linear term by a function. In $(1+1)$ dimensions this appears as

$m\omega x\to W(x)$0

or, in the equivalent $m\omega x\to W(x)$1 notation,

$m\omega x\to W(x)$2

For real $m\omega x\to W(x)$3, the Hamiltonian

$m\omega x\to W(x)$4

is Hermitian; for complex $m\omega x\to W(x)$5, the same formal structure supports pseudo-Hermitian and $m\omega x\to W(x)$6-symmetric realizations (Dutta et al., 2013).

The generalized formulation is not confined to one dimension. In $m\omega x\to W(x)$7 dimensions one finds radial substitutions such as

$m\omega x\to W(x)$8

and in curved backgrounds the ordinary replacement

$m\omega x\to W(x)$9

is extended to

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$0

with the cosmic-string literature often activating only the radial component $$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$1 (Deng et al., 2018). In the recent three-dimensional DSR treatment, the undeformed spatial operator remains the ordinary Dirac oscillator, while the deformation acts on the energy reconstruction rather than on the oscillator eigenfunctions themselves (Boumali et al., 25 Feb 2026).

2. Supersymmetric factorization and solvable sectors

A central structural fact is that the generalized Dirac oscillator decouples into supersymmetric partner equations. In the local $$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$2-dimensional case, one introduces first-order operators

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$3

or equivalently

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$4

and obtains

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$5

with partner potentials

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$6

The two spinor components satisfy Schrödinger-type equations with the same positive-$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$7 spectrum, differing only possibly in the zero mode. In exact SUSY, the zero mode satisfies

$$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$8

and the corresponding Dirac state has $$\bigl(c\sigma_x(p_x-iW(x)\sigma_z)+mc^2\sigma_z\bigr)\psi=E\psi,$$9, so the ground state is separable between spin and spatial degree of freedom (Laba et al., 5 Jul 2026).

This factorization is the basis of exact solvability. The DSR–Morse analysis recalls the standard shape-invariance condition

$f(x)$0

from which

$f(x)$1

follows algebraically. For the pseudo-Hermitian complexified Morse interaction

$f(x)$2

the partner potentials are shape invariant and the spatial spectrum is

$f(x)$3

so the Morse sector has only finitely many bound states (Boumali, 3 Mar 2026).

A distinct exactly solvable direction is the isotonic generalization. In $f(x)$4 dimensions, choosing

$f(x)$5

yields an effective isotonic potential, and the exact relativistic spectrum is

$f(x)$6

In the non-relativistic limit this reduces to an isotonic oscillator Hamiltonian on the half-line. In $f(x)$7 dimensions, the analogous choice

$f(x)$8

produces a radial isotonic equation and an anti-Jaynes–Cummings-like Hamiltonian in which the spin operators couple with the supercharges (Ghosh et al., 22 Feb 2025).

3. Pseudo-Hermiticity, $f(x)$9 symmetry, and complex interactions

Generalized Dirac oscillators with complex interactions form a major branch of the subject. The basic criterion is $H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$0-pseudo-Hermiticity,

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$1

with the translation-type metric

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$2

Because $H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$3 implements an imaginary shift,

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$4

the condition

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$5

guarantees pseudo-Hermiticity of the local GDO. For such systems one may define $H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$6 and obtain a Hermitian counterpart $H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$7. The 2013 analysis gave explicit complex Morse-type and periodic Rosen–Morse-type interactions satisfying this shifted-conjugation condition and showed that the non-Hermitian GDO can be isospectral to a Hermitian Dirac Hamiltonian with a real interaction function (Dutta et al., 2013).

The 2026 DSR review extends this framework by distinguishing the ordinary adjoint from the metric adjoint

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$8

so that the supersymmetric partner Hamiltonians become

$H_{\text{GDO}}=\sigma_x p+\sigma_y f(x)+\sigma_z m$9

The same paper also formulates the $c=1$0-symmetry criterion

$c=1$1

for the local partner Schrödinger operators and emphasizes that pseudo-Hermiticity or $c=1$2 symmetry secures a real spatial spectrum $c=1$3, while DSR modifies only the final map $c=1$4 (Boumali, 3 Mar 2026).

These complex extensions also admit model correspondences. The generalized Dirac oscillator is exactly identified with a generalized anti-Jaynes–Cummings Hamiltonian

$c=1$5

under the identification $c=1$6, $c=1$7. Under the sign reversal $c=1$8, which interchanges $c=1$9, the model becomes a generalized Jaynes–Cummings system. This places the GDO inside a broader algebraic family of spin-boson-like models (Dutta et al., 2013).

4. External fields, position-dependent mass, and curved geometry

One widely studied extension couples the generalized Dirac oscillator to a scalar electric potential. In $W(x)=m\omega x$0 dimensions,

$W(x)=m\omega x$1

and for the special choice

$W(x)=m\omega x$2

the second-order equation reduces to a SUSY form with the energy-dependent superpotential

$W(x)=m\omega x$3

This yields exact solutions whenever $W(x)=m\omega x$4 is shape invariant. The same analysis shows that sufficiently strong electric fields destroy the bounded eigenstates: bound states require

$W(x)=m\omega x$5

whereas for $W(x)=m\omega x$6 the effective confining term changes sign and discrete states disappear (Laba et al., 2018).

A related extension introduces a position-dependent mass. The $W(x)=m\omega x$7-dimensional equation

$W(x)=m\omega x$8

becomes exactly reducible when

$W(x)=m\omega x$9

The resulting scalar equation has supersymmetric form with

$f(x)=m\omega x$0

and bound states exist only below the critical electric-field strength

$f(x)=m\omega x$1

The same paper also constructs exact zero-energy states, including a step-profile solution that reduces to the Jackiw–Rebbi mode when the oscillator term vanishes (Ho et al., 2018).

Curved-space realizations are likewise standard. In cosmic-string space-time,

$f(x)=m\omega x$2

the conical defect enters through the effective angular parameter

$f(x)=m\omega x$3

and radial couplings such as the Cornell choice

$f(x)=m\omega x$4

lead to confluent-hypergeometric solutions, whereas Yukawa-, Hulthén-, generalized-Morse-, and singular-type couplings lead to hypergeometric or power-series solutions (Deng et al., 2018). In $f(x)=m\omega x$5-dimensional cosmic-string space-time with Aharonov–Casher coupling, the Coulomb-type generalized oscillator

$f(x)=m\omega x$6

produces exact radial Laguerre solutions, and the relativistic energy levels depend explicitly on the Coulomb strength $f(x)=m\omega x$7, the AC phase, the AC frequency, and the deficit parameter $f(x)=m\omega x$8; the degeneracy is broken by the Coulomb-strength parameter under the combined influence of curvature and the Aharonov–Casher effect (Chen et al., 2020).

5. Nonlocal kernels and deformed relativistic kinematics

A recent development replaces the local multiplicative interaction by a nonlocal integral operator

$f(x)=m\omega x$9

The resulting nonlocal generalized Dirac oscillator preserves factorization with

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$0

and the two spinor components satisfy nonlocal Schrödinger-type equations with partner kernels

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$1

where

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$2

The same work extends pseudo-Hermiticity to the kernel level through

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$3

and adapts current-based localization to obtain energy-dependent equivalent local potentials together with multiplicative Perey damping factors. The localization breaks down at current zeros, which diagnose spurious solutions of the nonlocal Schrödinger problem (Boumali, 5 Mar 2026).

Another major direction is DSR. In one dimension, the spatial spectral problem is first solved as

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$4

and only afterward is the relativistic energy reconstructed. In the undeformed theory,

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$5

but in the Magueijo–Smolin prescription the map becomes

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$6

whereas in the Amelino–Camelia prescription

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$7

The AC relation imposes the admissibility condition

$\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$8

so DSR can remove otherwise acceptable bound states solely through the deformed energy map; in the massless limit, the MS deformation collapses to the undeformed relation while the AC deformation remains nontrivial (Boumali, 3 Mar 2026).

The same logic extends to the three-dimensional modified Dirac oscillator. There the bound-state eigenfunctions retain the oscillator-spinor structure dictated by spherical symmetry, while DSR deforms the algebraic relation between $\mathbf p\to \mathbf p-i m\omega \beta \mathbf r,$9 and the relativistic energy. In the generalized first-order Planck-length expansion, the master equation becomes

$(1+1)$0

with

$(1+1)$1

This shows directly that the deformation signal increases with excitation through the oscillator scale and the spin–orbit splitting (Boumali et al., 25 Feb 2026).

6. Entanglement, equivalent models, and conceptual significance

The most recent conceptual extension treats the generalized Dirac oscillator as a bipartite relativistic system with Hilbert-space factorization

$(1+1)$2

Because the upper and lower spinor components are SUSY partners, the spin amplitudes are fixed algebraically by the relativistic energy: $(1+1)$3 For excited states,

$(1+1)$4

and the reduced spin density matrix has eigenvalues

$(1+1)$5

where

$(1+1)$6

The von Neumann entropy

$(1+1)$7

therefore depends only on the relativistic energy $(1+1)$8 and the SUSY-partner overlap $(1+1)$9. For odd superpotentials, $m\omega x\to W(x)$00, one has $m\omega x\to W(x)$01, the reduced density matrix is diagonal, the entanglement vanishes in the nonrelativistic limit, and it reaches the maximal qubit value $m\omega x\to W(x)$02 as $m\omega x\to W(x)$03 (Laba et al., 5 Jul 2026).

This entanglement result sharpens a broader pattern in the GDO literature. Hidden supersymmetric structure is not only a spectral device; it also controls reduced states, model equivalences, and effective descriptions. The generalized Dirac oscillator has been mapped to generalized anti-Jaynes–Cummings and Jaynes–Cummings Hamiltonians, to isotonic and Morse families, to magnetic-field problems in two dimensions through the exact identity

$m\omega x\to W(x)$04

and, in the nonlocal case, to equivalent local problems with energy-dependent potentials and damping factors (Dutta et al., 2013, Schulze-Halberg et al., 2021, Boumali, 5 Mar 2026).

Taken together, these developments define the generalized Dirac oscillator less as a single potential than as a framework. Its unifying feature is the replacement of the linear Dirac-oscillator coupling by a structured interaction—local or nonlocal, real or complex, flat-space or curved-space, undeformed or DSR-deformed—while preserving enough first-order factorization to make relativistic spectra, spinor structure, and, in some cases, quantum-information observables accessible in closed form.