⬇ Download Markdown · ⬇ Download LaTeX · ⬇ original · view original · CSV: table 1
Physica Scripta You may also like PAPER
Hermitian Hamiltonians David B Fairlie and Jean Nuyts - SUSY approach to Pauli Hamiltonians with an axial symmetry M V Ioffe, Kuru, J Negro et al.
View the article online for updates and enhancements.
This content was downloaded from IP address 103.191.84.79 on 23/05/2026 at 08:16 Phys. Scr. 101 (2026) 215101 https://doi.org/10.1088/1402-4896/ae6c6b aaaaaaaaaaaa- PAPER a RECEIVED 23 March 2026 REVISED
4 May 2026 ACCEPTED FOR PUBLICATION M Salazar–Ramírez
1,∗ aa , B C Lütfüoğlu 2 aa and H Bouguerne 3 aa 12 May 2026 1 PUBLISHED 21 May 2026 Escuela Superior de Cómputo, Instituto Politécnico Nacional, Av. Juan de Dios Bátiz esq. Av. Miguel Othón de Mendizábal, Col. Lindavista, Alc. Gustavo A. Madero, C.P. 07738, Ciudad de México, Mexico 2 Department of Physics, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czechia 3 Laboratoire de systèmes dynamiques et contrôle (L.S.D.C), Département des sciences de la matière, Faculté des Sciences Exactes et SNV,
Université de Oum-El-Bouaghi, 04000, Oum El Bouaghi, Algeria ∗ Author to whom any correspondence should be addressed.
E-mail: [email protected], [email protected] and [email protected]
Keywords: Dunkl-Pauli equation, Dunkl operators, Pauli equation, su(1,1) algebra, coherent states, magnetic field Abstract We investigate the two-dimensional Dunkl–Pauli equation for a spin–1/2 particle in the presence of an external magnetic field within an algebraic framework. While previous studies have provided exact solutions of the Dunkl–Pauli system, its underlying dynamical symmetry and algebraic structure remain largely unexplored. By employing Schrödinger factorization, we construct the generators of the su ( 1, 1 ) algebra and establish the associated symmetry of the radial sector. The energy spectrum is derived using irreducible unitary representations, and the corresponding Sturmian radial basis is obtained analytically. Furthermore, SU(1, 1) coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial breathing behavior. The results show that the Dunkl deformation introduces parity-dependent modifications in the spatial structure of the system while preserving its underlying algebraic dynamics.
The extension of quantum mechanics beyond the standard canonical framework has attracted considerable attention, particularly in the presence of discrete symmetries such as reflections. Early developments in this direction can be traced back to Wigner’s proposal of modified commutation relations [1] and Yang’s construction of momentum operators incorporating reflection symmetries in the harmonic oscillator [2]. These ideas were later explored within deformed Heisenberg algebras, where reflection-deformed structures appeared in generalized quantization schemes leading to parafields and parastatistics [3, 4], and were applied to systems such as the Calogero model [5], bosonization of supersymmetric quantum mechanics [6], anyonic systems [7], and more generally to deformations associated with generalized statistics [8].
A concrete operator realization of these concepts was established through the introduction of Dunkl operators, originally proposed by Dunkl [9], which combine differential and reflection operators into a unified differential–difference structure. Within this formalism, the canonical momentum is modified through the replacement of spatial derivatives by Dunkl operators, leading to a consistent incorporation of discrete symmetries at the dynamical level. This construction forms the basis of Wigner–Dunkl quantum mechanics [10], where the underlying algebra is deformed and the resulting dynamics depends explicitly on parity sectors [11–15]. This operator-based formulation provides a natural framework for analyzing quantum systems in which symmetry and solvability are deeply intertwined, and has led to a broad range of exactly solvable models and applications [16–20], including Dunkl oscillator systems and their nonrelativistic and relativistic extensions within algebraic frameworks [21–24].
The applicability of the Dunkl formalism extends to relativistic quantum systems, where reflection operators lead to modifications of both spectral and dynamical properties. In particular, relativistic wave equations such as the Klein–Gordon equation have been formulated within the Dunkl framework, yielding exact © 2026 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved.
Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al analytical solutions for oscillator and Coulomb-type interactions [25]. Relativistic Dunkl oscillators have also been investigated in various settings, including Dirac-type systems in the presence of external magnetic fields, providing further insight into the structure of deformed wave equations [26]. In addition, thermodynamic aspects of relativistic Dunkl systems have been explored, where Klein–Gordon and Dirac oscillator formulations can be treated within a unified framework, allowing the evaluation of physical quantities such as mean energy, entropy, and specific heat [27, 28]. Furthermore, algebraic approaches based on su ( 1, 1 ) symmetry have been employed to obtain exact spectra, eigenfunctions, and coherent states, highlighting the role of symmetry methods in the analysis of relativistic Dunkl systems [29].
In nonrelativistic quantum mechanics, Dunkl operators allow the exact or approximate treatment of a broad class of potentials, including harmonic [30, 31], pseudoharmonic [32], Coulomb [33, 34], Pöschl–Teller [35], Morse [36], and more general anharmonic interactions [37]. These studies show that the deformation parameters modify the spectral structure and influence physical observables through the presence of reflection operators.
An important extension of this framework arises in spin-dependent systems, where the coupling between spin and external magnetic fields introduces additional degrees of freedom. Within this context, the Dunkl–Pauli equation has been formulated and solved for spin–1/2 particles, yielding parity-dependent energy spectra and spinor eigenfunctions [38]. Time-dependent generalizations, such as the Dunkl–Pauli oscillator, further show that the inclusion of Dunkl derivatives significantly modifies the dynamical evolution of quantum states [39]. Recent studies have also explored thermodynamic and time-dependent aspects of Dunkl– Pauli systems in the presence of Aharonov–Bohm flux, highlighting the continued development of this framework [40, 41].
Despite these developments, the algebraic structure underlying the Dunkl–Pauli system has not been fully characterized. In particular, while exact solutions are available, an explicit construction of the associated su ( 1, 1 ) dynamical symmetry, together with the derivation of the corresponding Sturmian basis and coherent states, remains largely unexplored from an algebraic perspective. Without this algebraic framework, the organization of the spectrum and the description of quantum-state dynamics cannot be systematically addressed.
In this work, we investigate the two-dimensional Dunkl–Pauli equation for a spin–1/2 particle in a uniform magnetic field from an algebraic perspective. By employing Schrödinger factorization, we construct the generators of the su ( 1, 1 ) algebra and establish the corresponding dynamical symmetry of the system. This framework enables us to derive the energy spectrum and eigenfunctions, as well as to obtain the associated Sturmian radial basis in a systematic manner. We further construct the SU(1, 1) coherent states and analyze their time evolution. Our approach provides a unified algebraic formulation of the Dunkl–Pauli system and elucidates the role of reflection symmetry in shaping its spectral and dynamical properties.
The su ( 1, 1 ) algebra encodes the infinitesimal structure of the system and underlies the Lie group SU(1, 1) through exponentiation, yielding finite transformations. Within this framework, the algebra provides a natural setting for the construction of ladder operators and the factorization of the Hamiltonian, while the associated group plays a central role in the formulation of coherent states via displacement operators and in their time evolution [42, 43].
This paper is organized as follows. In section 2, we present the theoretical framework of the two-dimensional Dunkl–Pauli equation. In sections 3 and 4, we develop the su ( 1, 1 ) algebraic treatment and obtain the corresponding physical quantities, together with the associated graphical analysis, for the case of symmetric Dunkl parameters (|ν 1 | = |ν 2 |). Subsequently, in section 5, we extend this analysis to the case of asymmetric Dunkl parameters (|ν 1 | ≠ |ν 2 |) by presenting the corresponding plots and comparing both behaviors. Finally, in section 6, we summarize the main results and discuss their physical implications.
The stationary Pauli equation, obtained from the nonrelativistic limit of the Dirac theory while retaining the spin–magnetic interaction, can be written as [38]
e where the conjugate momenta include the electromagnetic coupling through
j = p j A . In the Dunkl c j framework, the canonical momenta are replaced by differential-reflection operators, introducing a discrete symmetry into the kinematic structure [9]. They are defined as
p j = iD j , 2
( 1 R j ) , ( 2 )
Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al here, D j denotes the Dunkl differential–reflection operator, ν j are the deformation parameters, and R j represent reflection operators associated with the coordinate x j . The reflection operators introduce a parity-dependent structure in the system, meaning that the dynamics explicitly depends on the behavior of the wavefunction under spatial inversion.
It is worth emphasizing that the operators defined in equation (2) correspond to the reflection group W = Z 2 × Z 2 , associated with independent reflections in each spatial coordinate. In a more general setting, Dunkl operators can be constructed for arbitrary finite reflection groups W, leading to richer algebraic structures [44, 45]. Their construction relies on operators satisfying
This structure is characteristic of Dunkl–type algebras and may be viewed as the Dunkl representation of a rational Cherednik algebra associated with the reflection group W. Unlike a simple Heisenberg deformation, the inclusion of reflection operators introduces parity-dependent terms that enhance the symmetry structure of the system [46–48].
The Wigner parameters are assumed to satisfy ν j > − 1/2 which guarantees a well-defined weighted Hilbert space H ( R 2 , x 1 2 1 x 2 2 2 dx 1 dx 2 ) and the integrability of the associated Dunkl measure. In this framework, the scalar product is naturally defined with the weighted measure x 1 2 1 x 2 2 2 dx 1 dx 2 , under which the Dunkl momentum operators remain Hermitian and the reflection structure of the system is consistently incorporated [10, 49]. A uniform external magnetic field is introduced through the symmetric gauge choice B A =
2 ( x 2 i ˆ + x 1 j ˆ) , under which the Dunkl–Pauli Hamiltonian takes the form [38]
g s µ B ( 1 + 1 R 1 + 2 R 2 ) B · S , ( 5 )
where ω c = eB/(mc) and μ B = |e|/(2mc) denote the cyclotron frequency and the Bohr magneton, respectively, while B represents the external magnetic field strength. The spin operator is given by S = σ/2 and g s = 2.0023 is the free electron g-factor. The operators B θ and J θ represent the angular contributions within the Dunkl framework.
Thus, the angular dependence of the Dunkl deformation is encoded in the operators [38]
B = 1
2 2 + (
1 tan
J = i [ + 1 2 cot ) + 2 cot ( 1 R 2 )
2 cos 2 ( 1 1 tan R 1 ) + ( 1 2 2 sin 2 ( 1 R 2 ) , ( 6 )
R 1 )] . ( 7 ) Because J commutes with R 1 R 2 , the Hilbert space splits into reflection sectors, and the wavefunction separates as [16, 18]
and the angular factor satisfies the eigenvalue equation
here λ ε is the eigenvalue of Dunkl-angular operator. For ε = + 1, this case corresponds to the situation in which either ε 1 = ε 2 = + 1 or ε 1 = ε 2 = − 1 (eigenvalues of the reflection operators R 1 and R 2 , respectively); therefore, the angular solutions take the form [38]
2 cos with θ ∈ [0, π] (x ∈ [ − 2, 2]), and ℓ corresponds to the angular momentum of the system. By identifying the radial function in the separated wavefunction as F m s ( r ) = F m ( s 1 , 2 ) ( r ) , the radial component of the Dunkl–Pauli equation assumes the general form [38] Here, P ( , 1 ) ( x ) denote Jacobi polynomials, obtained via the change of variable x = d 2 1 + 2 1 + 2 2 d + 2 dr r dr 2 + r 2 m 2 c 2 2 r
4
m c + + mB µ B g s m s ( 1 + 1 1 + 2 2 ) + 2 mE
( 12 ) The radial equation shows that the Dunkl parameters ν 1 and ν 2 modify the effective centrifugal term and the spin-magnetic interaction, thereby affecting both the localization and energy spectrum of the system.
3 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
The use of Schrödinger factorization allows us to reveal an underlying algebraic structure, which provides a systematic way to obtain the energy spectrum and eigenfunctions. By introducing the dimensionless radial variable = m c r , the radial equation (12) can be cast into the dimensionless form 2
d 2 1 + 2 1 + 2 2 d + 2 d d 2 + 2 2 2 + + 2 B µ B g s m s ( 1 +
1 1 + 4 E 2 2 ) + c 1 , 2 , m s c ( ) = 0.
( 13 )
To eliminate the first-order derivative term appearing in the dimensionless radial equation, we introduce the
d 2 1 , 2 , m s
dd
( + 2 )
1 , 2 , m s ( + 1 ) d 2 ( ) , d 2 , d 2 + d ( 15 )
1 , 2 , m s ( ) . ( 16 )
Substituting these expressions into the radial equation (13) and simplifying the resulting terms, the first-order derivative contribution is completely removed.
Consequently, after multiplying by −ρ 2 , the equation for 1 ,, m 2 s ( ) assumes a Schrödinger–like form. In particular, for the sector ε 1 = ε 2 = 1, one obtains
2 d
2 d 2 + 4 1 +
4 2 + + ( 1 + 2 ) 2 + 2 + 2 B µ B g s m s (
1 + 1 + 2 ) 4 E c 2
c+ , + , m s ( ) = 0,
( 17 )
within the Schrödinger factorization formalism, a closed set of su ( 1, 1 ) generators is constructed by adopting the ansatz
, + , m s ( ) = G
, + , m s ( ) , ( 18 )
expanding the factorization ansatz in equation (18) and collecting terms with equal powers of ρ, one obtains the second–order differential equation
A direct term-by-term comparison between equation (19) and the Schrödinger–like equation (17) yields the following system of algebraic relations for the factorization parameters
c4 E , 2 2 ) , ( 20 ) c
resulting in the factorization parameters A , B , C , F , and G listed in table 1.
and G that appear implicitly in equation (18), obtained by expanding this equation and comparing it term by term with equation (17). In this case, the conditions correspond to ε = + 1 (ε 1 = ε 2 = 1). The separation N* . constant is given by + = ± 2 ( + 1 + 2 ) , with 4 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al Table 1. Schrödinger factorization parameters for the sector ε 1 = ε 2 = 1.
+1 + , + −1 + + , +
3 2 3 2 +1 + , + −1 + + , + It then follows that the equation satisfied by
1 2 1 2 2 + + ( 1 + 2 + + ( 1 + 2 ) 2 + ( + , + + 1 ) 2 2 ) 2 + ( + , + 1 2 1 2 1 )
2 , m s ( ) can be cast in the factorized form
These results allow us to construct two new operators
1 + , + = ± d + d 2 ± 1 d 2 + d 2 2 + (
1 + 2 2 ( 23 ) , here, + , + 0 + , + , m s ( ) = 1 4 2 + 2
, + , m s ( 2 ) , ( 24 )
, + where χ +,+ /2 corresponds to the eigenvalue associated with the operator + acting on the radial Sturmian
0 functions. Equations (23) and (24) are readily seen to define a closed realization of the su ( 1, 1 ) algebra (see appendix for additional details)
0 . ( 25 ) In order to extract the energy spectrum of the Dunkl–Pauli equation in the presence of a magnetic field, we make use of the unitary irreducible representations of the noncompact Lie algebra su ( 1, 1 ) [50, 51]
the quadratic Casimir operator C 2 satisfies the eigenvalue relation
the above equation determines the Bargmann index k associated with the present problem in the form
2 + (
1 + 2 )
the quantization condition follows immediately from the combined use of equations (24), (26), and (28)
1 + 1 + 2 ) + c 2 E = 2 ( n
r + ) + ( 1 + 1 + 2 ) ,
( 29 ) c
from which the Dunkl–Pauli energy spectrum in a magnetic field is obtained
In the undeformed limit ν 1 = ν 2 = 0, this expression reduces to
, which corresponds to the ordinary Pauli–Landau spectrum obtained when the Dunkl deformation and reflection-dependent contributions vanish.
The differential equation that allows one to construct the wave function + ,, m + s ( ) corresponding to equation (17) is 5 ( 31 )
Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
1 2 x + 4 y + 4 n r + 2 + 2 2
y = 0, x 2 ( 32 )
which has the solution [52, 53]
A direct use of equations (32) and (33) allows one to construct the wave functions equation (17), which yields + , + , m s ( ) = = 2 n r ! ( n r + 2 + + ( 1 +
2 n r ! ( n r + 2 + 1 + 2 ) 2 + 1 )
e2 + 1 )
e2 / 2 2 / 2 m c
2 m c 2 1 + 1 + 2
2 1 + 1 + 2
2 + , + , m s ( ) corresponding to 2 1 2 + + (
1 + 2 ) + 2 L n
r2 2 + + (
1 + 2 ) ( 2 )
2 + 1 + 2 + 1 2 + 1 + 2 2 2 L
n ( ) , r
( 34 ) the irreducible unitary representation of the su ( 1, 1 ) Lie algebra for the Dunkl–Pauli equation in the presence of a magnetic field admits a Sturmian basis characterized by the group indices n and k, of the form
F n + r ,, + ( r ) = m c 2 m c 2 n r !
e 4 r ( n r + 2 k ) 2 k r 2 L n 2 r k 1 m c 2 r .
2 ( 35 )
The radial probability densities F n + r ,, + ( r ) 2 for ℓ = 1 and n r = 0, …, 5 are displayed in figures 1–3 for different deformation sectors (ν 1 , ν 2 ). In all cases, as n r increases, the principal maximum shifts toward larger radial 6 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
distances, i.e., to the right along the radial axis r, and an increasing number of nodes appears. A key observation is that the radial profiles depend solely on the combination ν 1 + ν 2 ; consequently, different pairs (ν 1 , ν 2 ) yielding the same sum produce identical radial probability densities.
The Dunkl deformation primarily modifies the position of these maxima: for ν 1 + ν 2 > 0, the curves move to larger values of r due to a stronger effective angular repulsion near the origin, whereas for ν 1 + ν 2 = 0 they remain close to the undeformed case. In contrast, when ν 1 + ν 2 < 0, the effective centrifugal contribution is reduced and the radial profile undergoes a global readjustment of its characteristic scale; however, the qualitative behavior under increasing n r remains unchanged, since the number of nodes grows and the principal maximum continues to shift toward larger radii. This behavior confirms that the effective parameter k governs the degree of radial confinement of the states.
3.1. SU(1, 1) radial coherent states and their time evolution We define the SU(1, 1) Perelomov coherent states as the action of the displacement operator on the lowest normalized state (see appendix for additional details)
( n + 2 k ) n ! ( 2 k )
1 / 2
n ( 36 ) k , n , applying the displacement operator to the ground radial state of the Dunkl oscillator, the coherent radial wave function can be written as R ( r , ) = 2 2 k 1 / 2 2 ( 1 ) m
c 2 r 4 exp ( 2 k )
m c 2 k r 2 1 2 n
L 2 k 1 n n = 0 m c 2 r . 2 ( 37 )
In order to evaluate the above series, we make use of the generating function of the Laguerre polynomials
1 exp y ) + 1 ( 1 xy 1 y , y < 1, ( 38 )
using this identity, the summation in equation (37) can be carried out in closed form, yielding
2 ) 2 k
) 4 k 1 / 2
r 2 k 1 2 exp m c r
2
2 + 1 ,
1 ( 39 )
finally, expressing the result in terms of the orbital quantum number ℓ together with the Wigner parameters ν 1 and ν 2 , we obtain the SU(1, 1) radial coherent states for the Dunkl–Pauli equation in the presence of a magnetic field in the form R ( r , ) = 2 ) 2 + 1 + 2 + 1 2 ( 1 ( 2 + 1 + 2 + 1 ) ( 1 ) 2 ( 2 + 1 + 2 + 1 )
1 / 2 m
c2 + 1 + 2 + 1 2
r 2 exp m c r
2
4 + 1 .
1
( 40 )
These coherent states describe semiclassical-like wave packets, whose evolution captures the dynamical behavior of the system in a compact analytical form.
7 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
The temporal evolution of these coherent states can be obtained in a direct way, since the third generator of 1 , + the algebra is proportional to the radial Hamiltonian, namely + =
2 H r . Consequently, the operator gov- 0 erning the time translation of the system is defined as (see appendix for additional details)
where t denotes an effective evolution parameter. Accordingly, the time-dependent Perelomov coherent state is introduced as
from the action of the time-translation operator on the lowest-weight state, one immediately finds
The similarity transformation of the ladder operators follows from the Baker–Campbell–Hausdorff formula together with the su ( 1, 1 ) commutation relations, yielding
As a consequence, the displacement operator evolves according to
2 it/ ℏ where the time-dependent parameter is defined as ξ(t) = ξ e ordered form of the displacement operator reads
D ( ( t )) = exp [ ( t ) + ] exp [ ] , ( 45 )
. In terms of this parameter, the normal0 ] exp [ * ( t ) ] , ( 46 )
compact form
with these results, the configuration-space representation of the time-dependent radial coherent state becomes
1 / 2 2 ) 2 + 1 + 2 + 1 2 ( 1 ( 2 + 1 + 2 + 1 ) ( 1 e 2 it / ) 2 ( 2 + 1 + 2 + 1 )
m c 2 + 1 + 2 + 1 2
r 2 exp m c r
2
4 e 2 it / + 1 , e 2 it / 1 < 1.
( 48 )
As the orbital quantum number ℓ increases (figures 4–8), the radial probability density of the coherent state moves away from the origin and progressively shifts toward larger radii. This behavior is characteristic of a coherent wave packet in the presence of an increasing centrifugal barrier: the factor r 4 ℓ suppresses the probability near r = 0, while the exponential term preserves the overall confinement, producing a ring-like distribution whose effective radius grows with ℓ.
Consequently, the coherent state evolves from being concentrated near the center to describing increasingly external radial orbits, analogous to the growth of the classical radius in a rotational oscillator. The overlapping 8 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
curves correspond to sectors (ν 1 , ν 2 ) with the same effective combination ν 1 + ν 2 , and therefore generate the same radial dynamics. Notably, all radial expressions depend only on the sum ν 1 + ν 2 ; consequently, pairs (ν 1 , ν 2 ) with the same sum yield identical curves.
9 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
Figures 9–11 display the time evolution of the radial probability density |R(r, ξ(t))| 2 for ℓ = 1. Within each panel, each curve corresponds to a fixed instant of time; thus, the dynamics is inferred by comparing the full temporal sequence. As t increases, the probability maximum first shifts toward larger values of r and, as the sequence continues, appears again at positions closer to the origin within the periodic cycle, revealing a radial ‘breathing’ motion. During this process, the profile alternates between narrower and higher shapes and broader and lower ones, reflecting the compression and expansion of the coherent packet.
Comparing different deformation sectors shows that the temporal pattern is common to all of them, changing only the radius around which the oscillation occurs. When ν 1 + ν 2 > 0, the entire evolution is shifted 10 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
toward larger radii; when ν 1 + ν 2 = 0, the curves remain practically coincident with the undeformed case; and when ν 1 + ν 2 < 0, the dynamics moves closer to the origin. Thus, the Dunkl deformation modifies the mean position of the coherent packet without altering the periodicity or the nature of its temporal evolution, highlighting that all radial expressions depend solely on the sum ν 1 + ν 2 .
In this section, we extend the algebraic treatment of the Dunkl–Pauli equation to the remaining reflection sectors. The analysis is carried out for the configurations (ε 1 , ε 2 ) ∈ {( − , − ), ( + , − ), ( − , + )}, which together with the case (ε 1 , ε 2 ) = ( + , + ) studied in section 3 correspond to ε = ± 1.
Although the radial equations differ among sectors, the underlying su ( 1, 1 ) algebraic structure remains the same. In particular, all cases share the same quadratic Casimir operator and Bargmann index, leading to identical Sturmian bases and coherent states. The differences arise only in the corresponding quantization conditions and energy spectra.
In what follows, we analyze each sector separately, emphasizing only the sector-dependent contributions while avoiding repetition of the algebraic construction.
In this sector, both reflection operators act with negative parity, which modifies the effective spin–magnetic coupling through the combination (1 − ν 1 − ν 2 ). The corresponding radial equation reads
2
r+ 2 1 + 2 1 + 2 2
r r 2 + r 2 m 2 c 2 2 r
4 +
( 49 )
After removing the first-order derivative term, the equation acquires a Schrödinger-like form
d 2 d 2 2 + 2 [
1 4 2 + + 2 2 B µ B g s m s
c( 1 1 2 ) + 4 E
c
The structure of the inverse-square term reflects the effective centrifugal barrier modified by the Dunkl parameters, while the constant terms encode the combined effect of the magnetic field and spin interaction.
By following a procedure analogous to that developed in section 3 for the ( + , + ) sector, as outlined in equations (19)–(25), one arrives at the quantization condition
c c 11 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al from which the energy spectrum follows as
( 52 ) .
This result shows that the Dunkl deformation modifies the Zeeman-type splitting through the factor (1 − ν 1 − ν 2 ), producing a sector-dependent shift in the energy levels.
In this mixed-parity sector, the reflection operators act asymmetrically, leading to a different effective coupling governed by (ν 1 − ν 2 ). The radial equation takes the form
2 d 2 1 + 2
1 + 2 2 d + 2 d d 2 + 4 1 2 + 2
2 2 B µ B g s m s (
1 + 2 ) + 1 c 4 E
c+ , , m s ( ) = 0,
( 53 )
with
In this sector, the angular quantum number ℓ takes half-integer values. This behavior originates from the asymmetric action of the reflection operators in the Dunkl angular equation, which modifies the admissible angular states and naturally leads to a semi-integer quantization of ℓ, analogous to fermionic angularmomentum structures [16, 18].
As a preliminary step toward the Schrödinger factorization procedure, we introduce the transformation defined in equation (14) in order to eliminate the first-order derivative term appearing in equation (53), namely, c
( 55 )
( 56 ) c and the corresponding energy spectrum is
( 57 ) .
Physically, this sector highlights how the asymmetry between the deformation parameters modifies the effective interaction, leading to distinct spectral shifts compared with the symmetric sectors.
This sector is complementary to the previous one, with the roles of ν 1 and ν 2 interchanged. The radial equation is
2 d 2 1 + 2
1 + 2 2 d + 2 d d 4 1 2 2 + 2
2 2 B µ B g s m s (
1 + 1 ) + 2 c 4 E , +
, m s ( c ) = 0.
( 58 )
After transformation,
d 2 d 2 c
( 59 )
( 60 ) c leading to the energy spectrum
12 2 + ( +
1 )( + 2 )
m s ( 1 + 2
2 1 ) . ( 61 )
Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
It is worth emphasizing that, in the undeformed limit ν 1 = ν 2 = 0, all reflection sectors (ε 1 , ε 2 ) recover the same energy spectrum given in equation (31).
This result confirms that the interchange ν 1 ↔ ν 2 produces a corresponding interchange in the spectral corrections, while preserving the same algebraic structure.
same deformation sectors considered in figures 1–3. The qualitative behavior with respect to n r remains the same: the number of nodes increases and the principal maximum moves toward larger r as the radial excitation grows. However, in comparison with the ℓ = 1 case, all distributions are shifted closer to the origin, i.e., toward smaller radial distances.
This difference arises from the reduction of the effective centrifugal barrier, since the Bargmann index
and the probability density becomes more localized at small radii, while the dependence on the deformation parameter ν 1 + ν 2 preserves the same trend observed in figures 1–3. This confirms the role of ℓ as an effective angular momentum governing the radial Dunkl dynamics.
Figures 15–19 display the radial coherent-state density |R(r, ξ)| 2 for half-integer orbital quantum numbers
term preserves the overall confinement. Moreover, some curves overlap because they depend only on the effective combination ν 1 + ν 2 . In contrast with figures 4–8, where integer values of ℓ produce more noticeable radial changes between consecutive panels, the present case shows a smoother and more continuous variation of the mean radius, highlighting the quasiclassical role of ℓ as an effective angular momentum controlling the orbital size of the coherent packet.
1 Figures 20–22 present the time evolution of |R(r, ξ(t))| 2 for =
2 , in contrast with figures 9–11 where ℓ = 1. In both cases, the same radial ‘breathing’ mode is observed, characterized by a periodic oscillation of the profile in which the probability maximum smoothly shifts within a temporal cycle. However, when passing 13 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
(0.4, − 0.4), ( − 0.4, 0.4), ( − 0.4, − 0.4).
( − 0.4, 0.4), ( − 0.4, − 0.4).
14 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
( − 0.4, 0.4), ( − 0.4, − 0.4).
(0.4, − 0.4), ( − 0.4, 0.4), ( − 0.4, − 0.4).
15 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
1 from ℓ = 1 to = 2 , the entire dynamics occurs at smaller radial distances due to the weakening of the effective centrifugal barrier associated with the orbital angular momentum. Although the periodicity and qualitative structure of the evolution remain unchanged, the mean radius around which the coherent packet oscillates depends explicitly on both ℓ and the effective combination ν 1 + ν 2 . This comparison confirms that ℓ plays the 16 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
role of an effective angular momentum controlling the orbital size of the coherent state without modifying the intrinsic nature of its temporal dynamics.
It is worth emphasizing that all results presented in sections 3 and 4 assume |ν 1 | = |ν 2 |, which guarantees that the radial dynamics depends exclusively on the sum ν 1 + ν 2 . This property, which leads to the degeneracy observed in the figures, will be broken in the asymmetric case |ν 1 | ≠ |ν 2 | analyzed in section 5.
For simplicity and compactness, only a few representative examples corresponding to the asymmetric Dunkl parameter configurations (|ν 1 | ≠ |ν 2 |) are presented, as illustrated in figure 23. The radial probability densities preserve the same qualitative behavior observed in the symmetric configurations: as the radial quantum number n r increases, the principal maximum shifts toward larger values of r and the number of nodes grows progressively.
The asymmetric deformations introduce only mild modifications in the amplitude and localization of the curves, without altering the overall structure of the profiles, mainly producing a gradual adjustment of the radial distribution. In contrast with the symmetric case (|ν 1 | = |ν 2 |), where all radial expressions depend solely on ν 1 + ν 2 , here the individual values of ν 1 and ν 2 become relevant, breaking the degeneracy observed in previous figures.
Several asymmetric configurations of the Dunkl parameters were analyzed. However, in order to avoid unnecessarily extending the present work, only two representative cases are included here, as shown in figure 24, since they adequately illustrate the main effects of the asymmetric Dunkl deformations on the radial
serve the same qualitative behavior observed in the symmetric case. As the radial quantum number n r increases, the principal maximum of the radial density shifts toward larger values of r and the number of nodes grows 17 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
4 progressively. The radial localization is mainly governed by the combination ν 1 + ν 2 , while the asymmetry between ν 1 and ν 2 introduces only mild modifications in the amplitude and shape of the profiles, without altering the overall structure of the radial distribution.
Although several values of ℓ were analyzed, only the representative cases ℓ = 1 and ℓ = 2 are included here, as illustrated in figures 25 and 26, in order to avoid making the manuscript unnecessarily more extensive. For these cases, the asymmetric configurations of the Dunkl parameters preserve the same qualitative behavior observed in the symmetric case: as ℓ increases, the radial distributions shift toward larger values of r and develop a more pronounced ring-like profile due to the increasing effective centrifugal barrier. However, unlike the symmetric case, where some curves may overlap, in the asymmetric configurations each pair (ν 1 , ν 2 ) generates a distinct radial profile, showing that the asymmetry removes the degeneracy present in the symmetric configurations.
The figures corresponding to the asymmetric case, shown in figure 27, display the time evolution of |R(r, ξ(t))| 2 for ℓ = 1. As in the symmetric case, a periodic oscillation of the radial profile is observed, in which the probability maximum smoothly shifts along the radial coordinate during the temporal cycle. The global structure and periodicity of the dynamics remain unchanged, confirming the robustness of the coherent behavior.
The differences between symmetric and asymmetric configurations manifest slightly in the amplitude and in the width of the distributions, which depend explicitly on the parameters (ν 1 , ν 2 ).
Although several half-integer values of ℓ were analyzed, only the representative cases = 12 and = 32 are presented here, as illustrated in figures 28 and 29, in order to keep the discussion compact. The radial coherent states for half-integer orbital quantum numbers and asymmetric Dunkl parameter configurations preserve the 18 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
same qualitative behavior observed in the symmetric case: as ℓ increases, the distributions shift toward larger values of r due to the growth of the effective centrifugal barrier. However, unlike the symmetric configurations, where several curves overlap because of their dependence on the combination ν 1 + ν 2 , in the asymmetric case 19 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
each pair (ν 1 , ν 2 ) generates a distinct radial profile. This shows that the asymmetry removes the degeneracy present in the symmetric case while preserving the overall radial dynamics.
1 The time evolution of |R(r, ξ(t))| 2 for fractional Dunkl parameters with =
2 , shown in figure 30, exhibits the same qualitative behavior as in the ℓ = 1 case: the radial profile evolves smoothly in time, with the probability maximum shifting within a finite radial region while preserving its shape and periodicity.
The main difference arises from the weaker effective centrifugal barrier for = 12 , which shifts the dynamics toward smaller radial distances and leads to higher peaks due to stronger radial localization. In both cases, fractional Dunkl parameters introduce only slight variations in amplitude and width, without altering the overall structure of the dynamics.
In contrast with the symmetric case (|ν 1 | = |ν 2 |), where all radial quantities depend solely on ν 1 + ν 2 , in the asymmetric case the individual values of ν 1 and ν 2 become relevant. Consequently, pairs with the same sum but different individual parameters yield distinct radial profiles, breaking the degeneracy observed in previous sections.
For ε = + 1 (i.e., ε 1 = ε 2 = 1 or ε 1 = ε 2 = − 1) and ε = − 1 (i.e., ε 1 = 1, ε 2 = − 1 or ε 1 = − 1, ε 2 = 1), the su ( 1, 1 ) algebraic framework yields identical radial Sturmian bases, since all sectors share the same quadratic Casimir invariant and Bargmann index k (see equation (27)). Consequently, the coherent states and their time evolution (see equations (40) and (48)) remain formally unchanged, with differences appearing only in the corresponding energy spectra.
Quantitative distinctions arise, however, in the numerical and graphical analysis due to the different allowed values of the orbital quantum number ℓ in each sector (see equations (11) and (54)). Although the su ( 1, 1 ) representation is formally the same, the integer versus half-integer character of ℓ modifies the effective centrifugal barrier and thus the radial localization scale. This accounts for the different radial ranges observed in the figures while preserving the same underlying dynamical structure.
As a consistency check of the present formalism, we consider the non-deformed limit corresponding to ν 1 = ν 2 = 0. In this regime, the expressions obtained for the radial wave functions, the energy spectrum, the Sturmian basis, and the coherent states together with their time evolution reduce exactly to the well-known results of the ordinary Pauli oscillator in the presence of a uniform magnetic field. This limiting behavior shows that the Dunkl parameters introduce a continuous deformation of the standard system while preserving the underlying SU(1, 1) algebraic structure, in agreement with results reported in the literature [50, 51, 54, 55].
In this work, we have developed an algebraic solution of the two-dimensional Dunkl–Pauli equation for a spin– 1/2 particle in the presence of a uniform magnetic field. The inclusion of Dunkl derivatives leads to reflectiondependent sectors and induces parity-sensitive quantum dynamics. We have shown that, for each reflection sector, the radial part of the Dunkl–Pauli equation admits an underlying su ( 1, 1 ) Lie algebraic structure.
By applying the Schrödinger factorization method, the generators of the su ( 1, 1 ) algebra were constructed explicitly, and the energy spectra were obtained through the theory of irreducible unitary representations. Although the energy spectra depend on the reflection parameters, all sectors are characterized by the same quadratic Casimir invariant and share an identical Bargmann index, which gives rise to a common Sturmian radial basis. Furthermore, this algebraic framework allows for the construction of SU(1, 1) Perelomov radial 20 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al coherent states and the analysis of their time evolution. These results highlight the unifying role of su ( 1, 1 ) symmetry in the algebraic description of the Dunkl–Pauli equation in a magnetic field.
The graphical analysis further elucidates the role of the Dunkl deformation and the orbital quantum number ℓ in both the stationary and time-dependent radial dynamics. The Sturmian densities show that the combined effect of ℓ and ν 1 + ν 2 directly controls the spatial localization of the states, shifting the probability distribution toward larger or smaller radii depending on the sign of ν 1 + ν 2 . In the coherent sector, the evolution exhibits a periodic radial ‘breathing’ mode whose mean radius depends on these same parameters, while the temporal frequency remains determined solely by the phase rotation of the coherent parameter ξ(t). Altogether, these results demonstrate that the Dunkl deformation modifies the spatial structure of the states without altering the intrinsic periodic character of the dynamics, reinforcing the organizing role of the underlying su ( 1, 1 ) symmetry in the Dunkl–Pauli system.
It is important to mention that an extension to non-reducible reflection groups W may affect the separability of the system, since such groups introduce couplings between spatial coordinates. In this case, it may become more difficult to reduce the problem to a simple radial equation as in the present formulation. Furthermore, the structure of the underlying symmetry algebra could also be modified, in the sense that the su ( 1, 1 ) symmetry may not be realized in the same form. These effects could, in turn, impact the construction of the Sturmian basis, coherent states, and their time evolution. However, a detailed assessment of these effects lies beyond the scope of the present work and is left for future investigation.
The extension of the Dunkl–Pauli equation in the presence of a magnetic field to higher dimensions appears feasible within the Dunkl framework, which is well established in arbitrary dimensions for scalar systems. However, the inclusion of spin degrees of freedom and the tensorial nature of the magnetic field introduce additional complexities that may affect separability and the underlying algebraic structure. Consequently, such a generalization requires a careful formulation and is left for future investigation.
B. C. L. is grateful to the Excellence project FoS UHK 2205/2025-2026 for the financial support. This work was partially supported by SNI–México, EDI–IPN, and SIP–IPN under project number 20260918.
All data that support the findings of this study are included within the article (and any supplementary files).
Lie algebras provide a powerful algebraic framework for the study of continuous symmetries in quantum systems. In this approach, the spectral properties of a physical model can be obtained through commutation relations and representation theory instead of solving the associated differential equations directly. In particular, the non-compact algebra su ( 1, 1 ) appears in a broad class of physical problems, ranging from relativistic quantum mechanics to quantum optics and squeezed states [56, 57]. The generators of the su ( 1, 1 ) algebra satisfy the commutation relations [50]
Introducing the ladder operators J ± = J 1 ± i J 2 , the commutation relations become
The Casimir operator is an invariant quantity that commutes with all generators of the algebra and therefore characterizes the irreducible representations, playing an essential role in the determination of the physical spectrum. For the su ( 1, 1 ) algebra, the quadratic Casimir operator is
The su ( 1, 1 ) algebra possesses different series of unitary representations, but here we consider only the discrete series. A basis for an irreducible representation is given by the set {|k, n〉, n = 0, 1, 2, …}, where k is the Bargmann index. The action of the generators on these basis states is 21 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al
n ( 2 k + n 1 ) k , n 1 ,
( A4 ) The state |k, 0〉 is the lowest normalized state, while the Casimir operator satisfies C 2 k , n = k ( k 1 ) k , n .
Perelomov coherent states extend the notion of standard coherent states to Lie groups through the action of a displacement operator on a reference state. From a physical point of view, these states describe semiclassical quantum configurations whose dynamics closely resembles the corresponding classical behavior.
For the non-compact group SU(1, 1), the coherent states |ζ〉 are defined through the displacement operator
Using the normal form of the displacement operator together with equation (A4), the Perelomov coherent states can be written as [51, 59]
( s + 2 k ) s ! ( 2 k )
s k , s . ( A8 )
When the Hamiltonian belongs to the Lie algebra, the time evolution operator reduces to a group element, and coherent states remain within the same family. This temporal stability transforms the Schrödinger equation into classical motion for the coherent parameters. Consider a Hamiltonian proportional to the diagonal generator [60, 61]
where ω is a real constant and ℏ = 1. Using the conjugation properties
the time-evolved coherent state becomes
Up to an irrelevant global phase, the evolved state remains a Perelomov coherent state whose parameter evolves according to
Thus, under a diagonal Hamiltonian, the coherent state rotates uniformly in the unit disk while preserving its functional form.
[1] Wigner E P 1950 Phys. Rev. 77 711 [2] Yang L 1951 Phys. Rev. 84 788 [3] Green H S 1953 Phys. Rev. 90 270 [4] Greenberg O W 1964 Phys. Rev. Lett. 13 598 [5] Brzezinski T, Egusquiza I L and Macfarlane A J 1993 Phys. Lett. B 311 202 [6] Plyushchay M S 1996 Ann. Phys. 245 339 [7] Plyushchay M S 1994 Phys. Lett. B 320 91 [8] Plyushchay M S 1997 Nucl. Phys. B 491 619 [9] Dunkl C F 1989 Trans. Amer. Math. Soc. 311 167 [10] Chung W S and Hassanabadi H 2019 One-dimensional quantum mechanics with Dunkl derivative Mod. Phys. Lett. A 34 1950190 [11] Dong S H, Chung W S, Junker G and Hassanabadi H 2022 Results Phys. 39 105664 [12] Junker G 2023 J. Phys. A 57 075201 [13] Chung W S, Junker G, Dong S H and Hassanabadi H 2023 EPL 141 32001 [14] Benchikha A, Hamil B, Lütfüoğlu B C and Khantoul B 2024 Phys. Scr. 99 105274 [15] Benchikha A, Hamil B and Lütfüoğlu B C 2025 Int. J. Geom. Methods Mod. Phys. 22 2550113 [16] Genest V X, Ismail M E H, Vinet L and Zhedanov A 2013 J. Phys. A: Math. Theor. 46 145201 [17] Genest V X, Vinet L and Zhedanov A 2013 J. Phys. A: Math. Theor. 46 325201 [18] Genest V X, Vinet L and Zhedanov A 2014 J. Phys. A: Math. Theor. 47 025203 [19] Quesne C 2024 EPL 145 62001 [20] Hamil B, Lütfüoğlu B C and Merad M 2025 Phys. Scr. 100 035301 [21] Salazar-Ramírez M, Ojeda-Guillén D, Mota R D and Granados V D 2017 Eur. Phys. J. Plus 132 39 22 Phys. Scr. 101 (2026) 215101 M Salazar–Ramírez et al [22] Salazar-Ramírez M, Ojeda-Guillén D and Mota R D 2018 Mod. Phys. Lett. A 33 1850112 [23] Mota R D, Ojeda-Guillén D, Salazar-Ramírez M and Granados V D 2021 Mod. Phys. Lett. A 36 2150066 [24] Ghazouani S 2021 Anal. Math. Phys. 11 35 [25] Hamil B and Lütfüoğlu B C 2022 Dunkl–Klein–Gordon equation in three-dimensions: the klein–gordon oscillator and coulomb
potential Few-Body Syst. 63 74 [26] Mota R D, Ojeda-Guillén D, Salazar-Ramírez M and Granados V D 2019 Ann. Phys. 411 167964 [27] Hamil B and Lütfüoğlu B C 2022 Eur. Phys. J. Plus 137 812 [28] Rouabhia N, Merad M and Hamil B 2023 EPL 143 52003 [29] Salazar-Ramírez M, Martínez-Nuño J A and Cordero-López M R 2026 SU(1, 1) Coherent states for the dunkl–klein–gordon equation
in its canonical form Few-Body Syst. 67 3 [30] Dong S H, Quezada L F, Chung W S, Sedaghatnia P and Hassanabadi H 2023 Ann. Phys. 451 169259 [31] Hamil B, Lütfüoğlu B C and Merad M 2025 Mod. Phys. Lett. A 40 2550094 [32] Mota R D and Ojeda-Guillén D 2022 Mod. Phys. Lett. A 37 2250006 [33] Genest V X, Lapointe A and Vinet L 2015 Phys. Lett. A 379 923 [34] Ghazouani S and Insaf S 2020 J. Phys. A: Math. Theor. 53 035202 [35] Schulze-Halberg A 2024 Approximate solutions of the Dunkl–Schrödinger equation for the hyperbolic Pöschl–Teller potential Few-
Body Syst 65 58 [36] Hamil B, Lütfüoğlu B C, Ikot A N and Okorie U S 2025 Spectral and thermal analysis of the morse potential within the dunkl
formalism: analytical approximations and applications Few-Body Syst. 66 27 [37] Schulze-Halberg A 2024 Mod. Phys. Lett. A 39 2450178 [38] Bouguerne H, Hamil B, Lütfüoğlu B C and Merad M 2024 Indian J. Phys. 98 4093 [39] Benchikha A, Hamil B and Lütfüoğlu B C 2025 Indian J. Phys. 99 4677 [40] Tedjani A and Khantoul B 2026 Thermodynamic properties of the dunkl-pauli oscillator in an aharonov-bohm flux Thermodynamic
Properties of the Dunkl-Pauli Oscillator in an Aharonov-Bohm Flux (https://arxiv.org/abs/2603.09364) 15 May 2026arXiv:2603. 09364 [41] Khantoul B and Tedjani A 2026 Time-dependent dunkl-pauli oscillator in the presence of the aharonov-bohm effect Physica Scripta
101 085204 [42] Wybourne B G 1974 Classical Groups for Physicists (Wiley) [43] Hall B C 2015 Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer) [44] Feigin M and Hakobyan T 2015 J. High Energy Phys. 11 107 [45] De Bie H, Oste R and Van der Jeugt J 2018 Ann. Phys. 389 192 [46] Etingof P and Ginzburg V 2002 Invent. Math. 147 243 [47] LePage E and Webster B 2023 Adv. Math. 433 109295 [48] Hakobyan T 2025 J. Phys. A 58 065201 [49] Junker G 2024 J. Phys. A 57 075201 [50] Barut A O and Girardello L 1971 Commun. Math. Phys. 21 41 [51] Perelomov A M 1986 Generalized Coherent States and Their Applications (Springer-Verlag) [52] Lebedev N N 1972 Special Functions and Their Applications (Dover Publications) [53] Gur Y and Mann A 2005 Phys. At. Nucl. 68 1700 [54] Rosas-Ortiz O and Frank A 2002 J. Phys. A: Math. Gen. 35 7101 [55] Ojeda-Guillén D, Mota R D and Granados V D 2012 J. Math. Phys. 53 102103 [56] Yeşiltaş Ö. 2015 Eur. Phys. J. Plus 130 128 [57] Gazeau J-P and del Olmo M A 2023 J. Opt. Soc. Am. B 40 1083 [58] Gerry C C 1985 Phys. Rev. A 31 2721 [59] Perelomov A M 1972 Commun. Math. Phys. 26 222 [60] Cohen-Tannoudji C, Diu B and Laloe F 1977 Quantum Mechanics (Wiley-VCH) [61] Gerry C C and Kiefer J 1988 Phys. Rev. A 37 665 23