Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation

Details Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation

2003-05-23

arxiv.org/abs/nlin/0305047v1

J.Math.Phys. 45 (2004) 4077-4105

Nonlinear Sciences

Exactly Solvable and Integrable Systems

41 pages, no figures

Scientific paper

10.1063/1.1780612

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 Schr\"{o}dinger 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, superintegrable or exactly solvable preserve these properties in the discrete case.

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