Dunkl-Deformed Pauli Equation

• The Dunkl-deformed Pauli equation is a quantum model that replaces standard derivatives with Dunkl derivatives, inherently incorporating reflection symmetry and parity dependence.
• It modifies kinetic, angular, and Zeeman coupling terms, leading to unique parity-dependent spectra and energy shifts controlled by deformation parameters ν1 and ν2.
• Exact solutions in static and time-dependent scenarios employ special functions like Jacobi and confluent hypergeometric functions, offering insights into quantum dynamics and thermodynamics.

Searching arXiv for relevant papers on the Dunkl-deformed Pauli equation and closely related extensions. arxiv_search(query="Dunkl Pauli equation magnetic field Aharonov-Bohm oscillator", max_results=10) The Dunkl-deformed Pauli equation is a deformation of the Pauli equation in which the ordinary momenta or derivatives are replaced by Dunkl momenta or Dunkl derivatives, thereby incorporating reflection operators directly into the quantum dynamics of spin-$\tfrac12$ systems. In two dimensions this replacement converts the standard Pauli problem into a reflection-symmetric differential-difference system with parity-dependent angular sectors, modified Zeeman coupling, and spectra that depend explicitly on the deformation parameters $\nu_1,\nu_2$. Exact realizations have been developed for a uniform external magnetic field, for Pauli oscillators, for Aharonov--Bohm flux backgrounds, and for time-dependent mass, frequency, and magnetic field profiles (Bouguerne et al., 2023, Benchikha et al., 2024, Khantoul et al., 6 Jan 2026, Tedjani et al., 10 Mar 2026).

1. Algebraic definition of the deformation

In its stationary form, the ordinary Pauli equation is written as

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$

with two-component spinor

$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$

The Dunkl deformation replaces the ordinary momentum by

$p_j=\frac{1}{i}D_j,$

where

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$

and the Wigner/Dunkl deformation parameters satisfy

$\nu_j>-\frac12.$

The reflection operators act as

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$

and obey the standard Dunkl relations

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$

The associated deformed Heisenberg algebra is

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$

or, equivalently in the magnetic-field formulation,

$\nu_1,\nu_2$0

This deformation is the defining structural change. It inserts reflection symmetry into the differential operator itself rather than appending parity as an external label. As a result, the Pauli equation becomes parity sensitive at the operator level, not merely at the level of boundary conditions or basis choice (Bouguerne et al., 2023, Benchikha et al., 2024).

2. Static Hamiltonian in a uniform magnetic field

For a two-dimensional nonrelativistic spin-$\nu_1,\nu_2$1 particle in a uniform magnetic field perpendicular to the plane, the symmetric gauge is taken as

$\nu_1,\nu_2$2

With this choice, the Hamiltonian is

$\nu_1,\nu_2$3

Using the Dunkl algebra, its orbital part becomes

$\nu_1,\nu_2$4

while the commutator term is

$\nu_1,\nu_2$5

Here the Dunkl Laplacian is

$\nu_1,\nu_2$6

Introducing

$\nu_1,\nu_2$7

the Hamiltonian is written as

$\nu_1,\nu_2$8

with

$\nu_1,\nu_2$9

In this form the deformation modifies three pieces simultaneously: the kinetic term through $$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$0, the angular coupling through Dunkl differential operators, and the Zeeman term through the reflection-dependent factor $$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$1. For $$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$2, the ordinary Pauli Hamiltonian is recovered (Bouguerne et al., 2023).

3. Polar reduction, angular operators, and parity sectors

In polar coordinates,

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$3

the Hamiltonian becomes

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$4

The Dunkl angular operators are

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$5

and

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$6

with the identity

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$7

The spinor is separated as

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$8

with

$$\frac{1}{2m}\left(\overrightarrow{\pi}\cdot\overrightarrow{\sigma}\right)^2\psi=E\psi, \qquad \pi_j=p_j-\frac{e}{c}A_j,$$9

The reflection operators act on the angular wavefunction by

$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$0

A further separation

$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$1

leads to the angular eigenvalue problem

$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$2

where

$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$3

are the reflection eigenvalues of $$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$4.

This is the basic parity decomposition of the Dunkl-deformed Pauli equation: the dynamics splits into sectors labeled by $$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$5, so parity dependence is not incidental but intrinsic to the spectral problem (Bouguerne et al., 2023, Khantoul et al., 6 Jan 2026).

Sector $$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$6	Reflection eigenvalues	Angular eigenvalue
$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$7	$$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$8 or $$\psi=\begin{pmatrix}\Phi\ \Psi\end{pmatrix}.$$9	$p_j=\frac{1}{i}D_j,$0
$p_j=\frac{1}{i}D_j,$1	$p_j=\frac{1}{i}D_j,$2 or $p_j=\frac{1}{i}D_j,$3	$p_j=\frac{1}{i}D_j,$4

For $p_j=\frac{1}{i}D_j,$5, $p_j=\frac{1}{i}D_j,$6. For $p_j=\frac{1}{i}D_j,$7, $p_j=\frac{1}{i}D_j,$8. The corresponding angular eigenfunctions are expressed through Jacobi polynomials in both the magnetic-field and Aharonov--Bohm formulations (Bouguerne et al., 2023, Tedjani et al., 10 Mar 2026).

4. Exact solutions and parity-dependent spectra

In the uniform-field problem, the angular part is completely controlled by Dunkl-reflection symmetry and is expressed through Jacobi polynomials, while the radial equation is a Dunkl-modified oscillator-type equation whose solutions are written in terms of the confluent hypergeometric function $p_j=\frac{1}{i}D_j,$9. Polynomial truncation $$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$0 yields discrete energies. For the four reflection sectors one obtains

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$1

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$2

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$3

and

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$4

The spectrum is therefore explicitly parity dependent and also depends on the deformation parameters $$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$5 (Bouguerne et al., 2023).

For the Dunkl-Pauli oscillator with Aharonov--Bohm flux, the angular part retains the Jacobi-polynomial structure, but the radial problem becomes a generalized oscillator with effective angular momenta

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$6

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$7

The inner and outer radial solutions are

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$8

$$D_j=\frac{\partial}{\partial x_j}+\frac{\nu_j}{x_j}(1-R_j),\qquad j=1,2,$$9

with

$\nu_j>-\frac12.$0

After regularization of the singular flux tube and matching at the solenoid radius, the $\nu_j>-\frac12.$1 limit gives

$\nu_j>-\frac12.$2

which yields the symmetry constraint

$\nu_j>-\frac12.$3

With this imposed, the physical outer spectrum is

$\nu_j>-\frac12.$4

and the full stationary state is

$\nu_j>-\frac12.$5

A central consequence is that the Aharonov--Bohm flux is not merely a gauge term: it introduces a singular interaction at the origin and enforces a compatibility condition that ties the reflection sectors to the Dunkl parameters (Tedjani et al., 10 Mar 2026).

5. Thermodynamic structure and Aharonov--Bohm effects

Once the exact spectrum is available, the Dunkl-deformed Pauli equation admits a closed canonical thermodynamics. In the uniform magnetic-field problem, for fixed angular momentum $\nu_j>-\frac12.$6, the partition function is defined by

$\nu_j>-\frac12.$7

and can be recast as

$\nu_j>-\frac12.$8

Here

$\nu_j>-\frac12.$9

and

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$0

The corresponding Helmholtz free energy, internal energy, heat capacity, and entropy are

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$1

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$2

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$3

and

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$4

The reported qualitative behavior is that the thermodynamic functions depend strongly on $$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$5 and parity, with the partition function increasing with temperature, the internal energy increasing monotonically, the heat capacity showing a peak near a critical temperature, and the entropy behaving differently at low temperature but tending to the standard result at high temperature (Bouguerne et al., 2023).

In the Aharonov--Bohm oscillator, the canonical partition function takes the compact form

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$6

with sector-dependent ground-state energy

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$7

The derived thermodynamic quantities are

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$8

$$R_jf(x_j)=f(-x_j),\qquad R_j^2=1,\qquad R_iR_j=R_jR_i,$$9

and

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$0

The heat capacity exhibits a Schottky-type anomaly controlled by the magnetic flux, while the high-temperature limit approaches the classical two-dimensional harmonic oscillator result $R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$1 (Tedjani et al., 10 Mar 2026).

A broader thermodynamic connection appears in the ideal Fermi-gas treatment in the Dunkl formalism. That work does not explicitly derive a Pauli equation, but it states that the Pauli exclusion principle is not removed, preserves fermionic $R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$2 occupancy, and concludes that magnetization decreases as the deformation parameter becomes more negative, so the Dunkl deformation suppresses spin alignment with the external magnetic field (Zenkhri et al., 15 Aug 2025).

6. Time-dependent generalizations and interpretive scope

The time-dependent Dunkl-Pauli oscillator extends the construction to nonstationary mass $R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$3, frequency $R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$4, and magnetic field $R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$5. In Cartesian coordinates the Hamiltonian is written as

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$6

and in polar form as

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$7

with

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$8

After separating the spin part, the polar Dunkl-Pauli equation becomes

$R_jx_i=-\delta_{ij}x_iR_j, \qquad R_jD_j=-D_jR_j.$9

A phase transformation,

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$0

removes the explicit $$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$1-linear term and reduces the problem to an effective Hamiltonian with time-dependent radial confinement. The resulting dynamics is solved by the Lewis--Riesenfeld invariant method, using an $$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$2 algebra and an auxiliary function $$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$3 or $$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$4 satisfying the Ermakov--Pinney equation

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$5

or, in the alternative notation,

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$6

The exact time-dependent wave functions retain the same special-function architecture as the stationary problem: Jacobi polynomials for the angular sector and associated Laguerre polynomials for the radial sector. One explicit form is

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$7

with

$$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$8

When Aharonov--Bohm flux is added to the time-dependent problem, the same flux-induced selection rule reappears: $$[x_i,D_j]=\delta_{ij}(1+2\nu_jR_j),\qquad [D_i,D_j]=[x_i,x_j]=0$$9 equivalently

$\nu_1,\nu_2$00

This means that topology and Dunkl reflection symmetry are not independent in the AB setting. The no-deformation limit $\nu_1,\nu_2$01 reproduces the ordinary Pauli or Pauli-oscillator models, the no-flux limit $\nu_1,\nu_2$02 removes the AB-induced selection rules, and the static limit of constant mass and frequency collapses the Lewis--Riesenfeld treatment to the stationary oscillator case (Benchikha et al., 2024, Khantoul et al., 6 Jan 2026).