# Umbral calculus, difference equations and the discrete Schrödinger equation

@article{Levi2004UmbralCD, title={Umbral calculus, difference equations and the discrete Schr{\"o}dinger equation}, author={D. Levi and P. Tempesta and P. Winternitz}, journal={Journal of Mathematical Physics}, year={2004}, volume={45}, pages={4077-4105} }

In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space–time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable… Expand

#### 38 Citations

Quantum mechanics and umbral calculus

- Mathematics, Physics
- 2008

In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrodinger equation substituting the… Expand

Discrete q-derivatives and symmetries of q-difference equations

- Mathematics, Physics
- 2003

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant… Expand

Heisenberg algebra, umbral calculus and orthogonal polynomials

- Physics, Mathematics
- 2008

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shall… Expand

Continuous symmetries of difference equations

- Mathematics, Physics
- 2005

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:… Expand

Heat Polynomials, Umbral Correspondence and Burgers Equations

- Physics, Mathematics
- 2006

We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the general… Expand

The discrete and periodic heat and harmonic oscillator equation

- Mathematics
- 2007

In this paper we present a generalization of the famous Dirac ladder formalism for the Schrödinger harmonic oscillator equation. Whereas the classical one-dimensional harmonic oscillator acts on… Expand

Discretization of partial differential equations preserving their physical symmetries

- Mathematics, Physics
- 2005

A procedure for obtaining a 'minimal' discretization of a partial differential equation, preserving all of its Lie point symmetries, is presented. 'Minimal' in this case means that the differential… Expand

Discretization of superintegrable systems on a plane

- Mathematics
- 2012

We construct difference analogues of so called Smorodinsky-Winternitz superintegrable systems in the Euclidean plane. Using methods of umbral calculus, we obtain difference equations for generalized… Expand

Multiple-scale analysis of dynamical systems on the lattice

- Mathematics
- 2011

We propose a new approach to the multiple-scale analysis of difference equations, in the context of the finite operator calculus. We derive the transformation formulae that map any given dynamical… Expand

Discretization of nonlinear evolution equations over associative function algebras

- Mathematics
- 2010

Abstract A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way that… Expand

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In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant… Expand

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