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