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Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation

Decio Levi·Piergiulio Tempesta·Pavel Winternitz
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Abstract

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 Schrödinger 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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Decio Levi, Piergiulio Tempesta, Pavel Winternitz. 2003-05-23. Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation. https://doi.org/10.1063/1.1780612

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