We generate solvable cases of the two angular equations resulting from variable separation in three-dimensional Dunkl-Schrödinger equation expressed spherical coordinates. It is shown that Dunkl formalism interrelates these with trigonometric Pöschl-Teller systems. Based on this interrelation, we use point transformations and Darboux-Crum to construct new equations. Instead stationary energy, c...

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax ope...

We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces polynomials bounded degree in the Euclidean unit ball, where weight is reflection-invariant form $(1-\lVert x \rVert^2)^\alpha \prod_{i=1}^d \lvert x_i \rvert^{\gamma_i}$, $\alpha, \gamma_1, \dots, \gamma_d > -1$. Said are measured Dunkl-Sobolev-type norms which same $\mathrm{L}^2$ norm used to contro...

Dunkl operators may be regarded as differential-difference parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for heat flows in $\mathbb{R}^d$ is introduced employing full "gradient" induced corresponding carr\'{e} du champ operator then $L^p$ boundedness studied all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt ...

On $$\mathbb {R}^N$$ equipped with a normalized root system R and multiplicity function $$k\ge 0$$ let us consider (not necessarily radial) kernel $$K({\mathbf {x}})$$ satisfying $$|\partial ^\beta K({\mathbf {x}})|\lesssim \Vert {\mathbf {x}}\Vert ^{-{\mathbf {N}}-|\beta |}$$ for $$|\beta |\le s$$ , where $${\mathbf {N}}$$ is the homogeneous dimension of $$({\mathbb {R}}^N,R,k)$$ . We addition...

We construct several new families of exactly and quasi-exactly solvable BCN -type Calogero– Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of BN type which, together with the original BN -type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of qua...

Analogous of Riesz potentials and Riesz transforms are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. The L boundedness of these operators is established in certain cases.