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

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 us to construct q-deformed Schrödinger equations in higher dimensions. The equivalence of these Schrödinger 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 suq(1|1)-representation.