Dunkl Number Coherent States

• Dunkl number coherent states are coherent states built on deformed number bases where reflection operators induce parity-sensitive modifications in ladder structures and factorials.
• They employ both Glauber-type eigenstate expansions and SU(1,1) Perelomov techniques to model quantum systems such as oscillators, Coulomb, and Kerr dynamics in weighted Hilbert spaces.
• The framework unifies diverse quantum models by incorporating modified commutation relations, parity-dependent spectra, and deformation parameters that adjust nonclassical observables.

Dunkl number coherent states are coherent-state constructions built on Dunkl-deformed number bases, i.e. on Fock-like or Sturmian bases whose ladder structure is modified by reflection operators and multiplicity parameters. In the one-dimensional $\mathbb{Z}_2$ setting they appear either as Glauber-type eigenstates of a Dunkl annihilation operator, expanded with deformed factorials $[n]_\mu!$, or as $SU(1,1)$ Perelomov number coherent states obtained by acting with a displacement operator on a Dunkl number state $|n\rangle$. Across oscillator, Coulomb, Kerr, parametric-amplifier, and Dunkl–Klein–Gordon models, the unifying ingredients are parity-sensitive Dunkl integers, weighted inner products, and $su(1,1)$ representations whose Bargmann indices depend on the Dunkl parameters (Ojeda-Guillén et al., 24 Apr 2026, Guillén et al., 18 Feb 2026, Salazar-Ramírez et al., 2016).

1. Dunkl-deformed number structure

The basic one-dimensional Dunkl operator is

$$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$

so the deformation couples differentiation to reflection. In the oscillator realization, the corresponding ladder operators act on Dunkl number states $|n\rangle$ through a parity-dependent integer

$[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$

with $[2m]_\mu=2m$ and $[2m+1]_\mu=2m+1+2\mu$. The associated Dunkl factorial $[n]_\mu!$0 replaces the ordinary factorial in coherent-state expansions, and the number operator satisfies $[n]_\mu!$1 in the sense inferred from the ladder action (Ojeda-Guillén et al., 24 Apr 2026).

This deformation is not merely a shift of the spectrum. Because $[n]_\mu!$2 distinguishes even and odd excitations, parity becomes intrinsic to the number basis itself. The commutation relations involve the reflection operator,

$[n]_\mu!$3

and the deformed inner product acquires the weight $[n]_\mu!$4. Consequently, Dunkl number states are orthonormal in a weighted Hilbert space rather than in the ordinary $[n]_\mu!$5 scalar product (Ojeda-Guillén et al., 24 Apr 2026).

In the broader harmonic-analysis formulation, Dunkl operators are attached to finite reflection groups and multiplicity functions $[n]_\mu!$6, and the Dunkl kernel $[n]_\mu!$7 replaces the ordinary exponential as the joint eigenfunction of the commuting Dunkl derivatives. This places Dunkl number coherent states within a general framework of reflection-deformed special functions, reproducing kernels, and weighted Gaussian Hilbert spaces (Dunkl, 2012).

2. Principal coherent-state constructions

A first construction is the Glauber-type Dunkl coherent state, defined by the eigenvalue equation

$[n]_\mu!$8

with expansion

$[n]_\mu!$9

This is the direct Dunkl analogue of the standard coherent-state expansion in number states, with $SU(1,1)$0 replaced by $SU(1,1)$1. These states minimize a generalized Heisenberg uncertainty relation, form an overcomplete set with respect to the weighted inner product, and reduce to ordinary Glauber coherent states when $SU(1,1)$2 (Ojeda-Guillén et al., 24 Apr 2026).

Because the coefficients depend on $SU(1,1)$3, the state naturally decomposes into even and odd ladders. The parity-resolved components

$SU(1,1)$4

are not an auxiliary refinement but part of the defining Dunkl structure. In models where the Hamiltonian commutes with the reflection operator, the even and odd sectors evolve independently, and coherent-state dynamics must be analyzed sector by sector (Ojeda-Guillén et al., 24 Apr 2026).

A second construction uses the $SU(1,1)$5 displacement operator. For the positive discrete series $SU(1,1)$6, Perelomov coherent states take the standard form

$SU(1,1)$7

In Dunkl oscillator and Dunkl-Coulomb problems, $SU(1,1)$8 is realized by a Sturmian radial basis whose representation parameter $SU(1,1)$9 is shifted by Dunkl parameters and reflection data. This yields closed-form radial coherent states in configuration space after summing the Laguerre-series expansion (Salazar-Ramírez et al., 2016, Salazar-Ramírez et al., 2016).

A third, more specific usage of the terminology appears in the Dunkl parametric amplifier, where the exact eigenstates are written as

$|n\rangle$0

These are Perelomov-type $|n\rangle$1 number coherent states built from arbitrary Dunkl number states rather than from the lowest-weight state alone. In this usage, “number coherent” emphasizes that the reference vector is $|n\rangle$2, while the underlying representation remains Dunkl-deformed through parity and the Bargmann indices $|n\rangle$3 (Guillén et al., 18 Feb 2026).

3. $|n\rangle$4 organization and parity sectors

Quadratic combinations of the Dunkl ladder operators define

$|n\rangle$5

and these satisfy the undeformed $|n\rangle$6 commutation relations

$|n\rangle$7

This is one of the central structural facts in the subject: the elementary oscillator algebra is reflection-deformed, but its quadratic closure is standard $|n\rangle$8 (Ojeda-Guillén et al., 24 Apr 2026).

The Casimir eigenvalue splits according to parity. In the one-dimensional Dunkl oscillator, the discrete-series Bargmann indices are

$|n\rangle$9

Hence even and odd excitations belong to different $su(1,1)$0 representations. This parity bifurcation is a persistent feature: it controls the spectral decomposition, the coherent-state coefficients, and the time evolution whenever the Hamiltonian preserves reflection symmetry (Ojeda-Guillén et al., 24 Apr 2026).

In higher-dimensional radial problems, the same mechanism reappears with more elaborate representation parameters. For the two-dimensional Dunkl oscillator, the radial sector realizes $su(1,1)$1 with

$su(1,1)$2

where $su(1,1)$3 is the angular quantum number. For the two-dimensional Dunkl-Coulomb problem, the Bargmann index becomes

$su(1,1)$4

Thus the “number” variable of the coherent state is no longer simply an occupation number; it is the radial excitation label of a Dunkl-modified $su(1,1)$5 representation (Salazar-Ramírez et al., 2016, Salazar-Ramírez et al., 2016).

The same representation-theoretic pattern extends to relativistic models. In the $su(1,1)$6-dimensional Dunkl–Klein–Gordon oscillator and Dunkl–Coulomb-like problem, Schrödinger factorization again produces $su(1,1)$7 generators, while the Bargmann index depends on the effective angular momentum $su(1,1)$8, the total Dunkl parameter $su(1,1)$9, the dimension $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$0, and, in the Coulomb case, the coupling $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$1. This shows that Dunkl number coherent states are best understood as coherent states of deformed representation spaces rather than as a single oscillator-specific construction (Salazar-Ramírez et al., 25 Jun 2026).

4. Dynamical behavior and nonclassical observables

The Dunkl Kerr oscillator provides the clearest example of how Dunkl number coherent states affect dynamics. Starting from an initial state that is a superposition of even and odd Dunkl coherent states, the exact parity-dependent spectrum leads to collapse and revival phenomena in the field quadrature and in the survival probability. The Dunkl parameter $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$2 modulates the fractional revivals, the fundamental revival period remains $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$3, and for $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$4 a perfect additional revival occurs at $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$5. The same analysis shows interference-induced squeezed states around $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$6, while the standard Kerr medium is recovered at $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$7 (Ojeda-Guillén et al., 24 Apr 2026).

In the Dunkl parametric amplifier, the Hamiltonian is diagonalized by both an $$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$8 tilting transformation and a generalized Bogoliubov transformation, and the exact eigenfunctions are Dunkl number coherent states. The spectrum is

$$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R), \qquad Rf(x)=f(-x),$$9

so the deformation enters through the representation rather than through the generalized Rabi frequency. The squeezed vacuum has a Mandel parameter independent of $|n\rangle$0, whereas the second-order correlation function $|n\rangle$1 depends explicitly on $|n\rangle$2, modifying photon bunching effects (Guillén et al., 18 Feb 2026).

For radial $|n\rangle$3 coherent states of the two-dimensional Dunkl oscillator, time evolution preserves the Perelomov form: the coherent parameter rotates as $|n\rangle$4, up to a global phase. This gives a strictly algebraic notion of temporal stability absent in generic nonlinear evolutions. In the higher-dimensional Dunkl–Klein–Gordon setting, the same mechanism produces time-dependent radial coherent states whose probability densities display a characteristic radial oscillation behavior, with the Dunkl deformation changing localization while leaving the underlying $|n\rangle$5 dynamics intact (Salazar-Ramírez et al., 2016, Salazar-Ramírez et al., 25 Jun 2026).

These examples also clarify a frequent misconception. The deformation parameter does not only shift energies; it changes phase accumulation differently in distinct parity ladders, alters the structure of coherent-state coefficients, and reshapes noise observables such as quadrature variances and intensity correlations. The nonclassical content is therefore encoded simultaneously in the algebra, in the representation labels, and in the parity decomposition.

5. Geometric, relativistic, and generalized extensions

The radial $|n\rangle$6 program extends beyond nonrelativistic oscillator models. In the two-dimensional Dunkl-Coulomb problem, coherent states are built on a Sturmian basis of Laguerre functions, and the resulting radial Perelomov states have a closed form after applying the tilting transformation back to the physical Hilbert space. The normalization is exact with respect to the Dunkl radial measure $|n\rangle$7, and the Dunkl parameters enter through the effective lowest weight $|n\rangle$8 (Salazar-Ramírez et al., 2016).

A relativistic variant appears in the canonical Dunkl–Klein–Gordon equation. In the even-parity sector and in the regime where the curvature constant $|n\rangle$9 is much smaller than the kinetic energy, Schrödinger factorization yields an $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$0 structure whose Bargmann index is fixed by the Dunkl parameter $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$1. The corresponding Perelomov coherent states are constructed in coordinate space and retain the standard group-theoretic form, although the spectrum is generally complex because the effective Hamiltonian is non-Hermitian. This identifies a relativistic, curved-space version of Dunkl number coherent states (Salazar-Ramírez et al., 15 Jul 2025).

The $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$2-dimensional Dunkl–Klein–Gordon oscillator and bound-state Dunkl–Coulomb-like problem show that the construction is not confined to $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$3 one-dimensional systems. Their radial coherent states remain $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$4 Perelomov states, but the representation parameter now encodes dimension, angular quantum numbers, and the sum of the Dunkl parameters. The deformation introduces parity-dependent modifications in the spatial structure while preserving the algebraic dynamics (Salazar-Ramírez et al., 25 Jun 2026).

Several adjacent lines of work provide natural, though not always fully realized, generalizations. In quasi-exactly solvable Wigner–Dunkl quantum mechanics, the extended Dunkl derivative $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$5 reorganizes anharmonic Hamiltonians into forms adapted to polynomial algebraic methods. This suggests nonlinear or truncated Dunkl number coherent states based on finite-dimensional invariant subspaces rather than on full Fock towers (Quesne, 2024). Likewise, the Dunkl–Darboux III oscillator on a space of nonconstant curvature yields exactly solvable spectra with reflection parameters and a position-dependent mass; this suggests Gazeau–Klauder or nonlinear coherent-state families controlled simultaneously by curvature $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$6 and Dunkl parameters $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$7 (Ballesteros et al., 2022).

At the level of harmonic analysis, the Dunkl kernel and intertwining operator offer the analytic infrastructure for coherent-state kernels on reflection-group backgrounds. For dihedral systems, explicit formulas for the Dunkl kernel, generalized Bessel functions, and integral representations make it plausible to define coherent states adapted to non-$[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$8 reflection symmetries, with overlaps governed by group-averaged Dunkl kernels rather than by ordinary exponentials (Deleaval et al., 2015).

6. Terminology, standard limits, and conceptual scope

In the cited literature, the expression “Dunkl number coherent states” does not denote a single universally fixed object. It refers to at least three closely related constructions. First, it may mean coherent states expanded in a Dunkl number basis with coefficients $[n]_\mu=n+\mu\bigl(1-(-1)^n\bigr),$9, as in the Dunkl Kerr oscillator. Second, it may mean $[2m]_\mu=2m$0 Perelomov number coherent states $[2m]_\mu=2m$1 built from arbitrary Dunkl number states, as in the Dunkl parametric amplifier. Third, it may be used more broadly for $[2m]_\mu=2m$2 coherent states whose representation label $[2m]_\mu=2m$3 is itself a Dunkl-modified quantum number, as in radial Dunkl oscillator, Coulomb, and Klein–Gordon systems (Ojeda-Guillén et al., 24 Apr 2026, Guillén et al., 18 Feb 2026, Salazar-Ramírez et al., 2016).

Despite this variation in usage, the limiting behavior is uniform. When the Dunkl deformation vanishes, $[2m]_\mu=2m$4, $[2m]_\mu=2m$5, $[2m]_\mu=2m$6, and the weighted scalar product reduces to the ordinary one. Accordingly, Dunkl coherent states become standard Glauber coherent states, Dunkl $[2m]_\mu=2m$7 states become the usual Perelomov or squeezed number states, the Kerr and parametric-amplifier spectra reduce to their undeformed forms, and the higher-dimensional radial constructions collapse to ordinary oscillator or Coulomb coherent states (Ojeda-Guillén et al., 24 Apr 2026, Guillén et al., 18 Feb 2026, Salazar-Ramírez et al., 2016, Salazar-Ramírez et al., 25 Jun 2026).

The decisive conceptual point is therefore not the existence of one preferred definition, but the persistence of a common structure across models: a deformed number basis, parity-sensitive ladder relations, $[2m]_\mu=2m$8 or oscillator-like coherent-state machinery, and analytic realizations in weighted Hilbert spaces. In that sense, Dunkl number coherent states are the coherent-state manifestations of reflection-deformed quantum kinematics.