q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials

TL;DR

This paper introduces a q-deformation framework for harmonic and Clifford analysis, defining q-Dirac and Laplace operators, and explores q-Hermite and Laguerre polynomials with applications to quantum equations in higher dimensions.

Contribution

It develops a novel q-deformation of differential operators and polynomials, connecting them to quantum algebra representations and higher-dimensional quantum equations.

Findings

Defined a q-Dirac operator inspired by q-derivatives.

Constructed q-deformed Schrödinger equations in multiple dimensions.

Connected q-Laguerre polynomials to su_q(1|1) representations.

Abstract

We define a q-deformation of the Dirac operator, inspired by the one dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows to construct q-deformed Schroedinger equations in higher dimensions. The equivalence of these Schroedinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford-Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal with respect to an m-dimensional q-integration, which is related to integration on q-Euclidean space. The q-Laguerre polynomials are the eigenvectors of an su_q(1|1)-representation.