Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials

Abstract

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat{P},\hat{M}]=1$. In ordinary quantum mechanics $\hat{P}$ is the derivative and $\hat{M}$ the coordinate operator. Here we shall realize $\hat{P}$ as a second order differential operator and $\hat{M}$ as a first order integral one. We show that this makes it possible to solve large classes of differential and integro-differential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing so called flatenned beams in laser theory

19 pages, 5 figures

References in corpus (6)

• Lie-algebraic discretization of differential equations
• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Representations of Monomiality Principle with Sheffer-type Polynomials and Boson Normal Ordering
• An Elementary Introduction to Groups and Representations
• Lorentz and Galilei Invariance on Lattices
• Canonical Commutation Relation Preserving Maps

Cited by in corpus (4)

• Direct spreading measures of Laguerre polynomials
• (Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries
• Operational vs. Umbral Methods and Borel Transform
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle