Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials

TL;DR

This paper explores the connection between umbral calculus and Heisenberg algebra, introducing new orthogonal polynomials via differential and integral operators, with applications in solving differential equations and laser beam modeling.

Contribution

It presents a novel realization of the Heisenberg algebra using differential and integral operators, leading to new classes of orthogonal polynomials related to Laguerre polynomials.

Findings

New classes of orthogonal polynomials introduced

Method for solving differential and integro-differential equations developed

Application to modeling flattened laser beams demonstrated

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