Narrow Results

Let ${{𝒮}}_N$ be symmetric group of degree $N$. In the representation of ${{𝒮}}_N$ corresponding to trapeziform partition (precisely $\tau =(m^{2k},a,b)$, $0\le b\le a<m$), we propose nonsymmetric Jack polynomials in $2mk+a+b$ variables which are singular at $\frac{1}{m+2}$. Moreover those nonsymmetric Jack polynomials span a module of ${{𝒮}}_N$ which is isomorphic to the representation of type $\sigma =(mk+a,mk+b)$.

We construct a quantum Hamiltonian for the isotonic Mathews–Lakshmanan oscillator system within the Dunkl formalism. It is shown that the Dunkl Hamiltonian does not admit bound states if the isotonic term is present. If the isotonic term is not present, then the Dunkl Hamiltonian admits bound states of even parity only. These results are in contrast to the findings for the conventional Mathews–Lakshmanan oscillator system.

We use a symmetrized Hamiltonian containing Dunkl operators to generate a Dunkl–Schrödinger equation with position-dependent mass. Our equation is shown to admit arbitrary-order Darboux transformations if the mass either has odd parity or is an inverse-square function. We construct the Darboux transformations and apply them to a Dunkl–Schrödinger system featuring a rational double well potential.

In this paper, we consider two specific quantum systems governed by the one-dimensional Klein–Gordon equation within the Dunkl formalism. Our systems are characterized by a symmetrized Coulomb potential and an inverse quadratic interaction, respectively. For both systems, we construct closed-form solutions of bound state type.

We consider the one-dimensional Schrödinger equation for the sextic anharmonic oscillator potential within the Dunkl formalism. Solutions of bound state type are constructed, and results are compared to the conventional scenario.

In this paper, we consider two fundamental problems in the framework of the Dunkl derivative: the Bosonic oscillator model of spin 0 under magnetic field: Landau levels and the Duffin–Kemmer–Petiau equation with the Coulomb potential. In both cases, we obtain the exact analytical solutions for the bound states in the general case for the different eigenvalues of the reflection operator.

In this paper, we introduce matrix operator algebra involving a universal curvature constant and using the Dunkl derivative. Consequently, the Dirac equation can be written without spin connections. Iterating the Dirac equation gives the Klein–Gordon equation in its canonical form without first-order Dunkl derivatives. This leads to a new form for the static metric based on the Dunkl operator in curved space.

We apply generalized Darboux transformations to Schrödinger equations at fixed energy that contain Dunkl derivatives, and are equipped with a position-dependent mass. Our Darboux transformations, originating from coupled Korteweg–De Vries systems, consist of two algorithms that generate new solutions, potentials and position-dependent mass functions for the Dunkl–Schrödinger system.

In this paper, we construct the general Hamiltonian for position-dependent mass systems within the Dunkl formalism. The associated Schrödinger equation is shown to admit closed-form solutions of bound state type for two particular Dunkl–Liénard systems, given by the Dunkl–Higgs oscillator, and a spiked mass Dunkl oscillator. Differences to the conventional scenario are discussed.

We construct closed-form solutions to the one-dimensional, stationary Dunkl–Schrödinger equation for the inverted oscillator potential with an inverse quadratic singularity at the origin. Conditions are derived for our solutions to represent bound states, and an example is given.

In this paper, we construct a deformed Schwarzschild black hole from the de Sitter gauge theory of gravity within Dunkl generalization and we determine the metric coefficients versus Dunkl parameter and parity operators. Since the spacetime coordinates are not affected by the group transformations, only fields are allowed to change under the action of the symmetry group. A particular ansatz for the gauge fields is chosen and the components of the strength tensor are computed as well. Additionally, we analyze the modifications on the thermodynamic properties to a spherically symmetric black hole due to Dunkl parameters for even and odd parities. Finally, we verify a novel remark highlighted from heat capacity: the appearance of a phase transition when the odd parity is taken into account.

In this paper, we introduce a Dunkl-type spherical maximal operator associated with a finite reflection group and study its boundedness. In view of this study, we give some results on suitable square functions and maximal multiplier operators in the Dunkl setting.