Umbral calculus, difference equations and the discrete Schrödinger equation

2003/05/23 by D. Levi, Decio Levi, Piergiulio Tempesta +2 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #hep-lat #hep-th #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1063/1.1780612

published as J.Math.Phys. 45 (2004) 4077-4105 · 41 pages, no figures

arxiv created 2003/05/23 · openalex publication_date 2004/10/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

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 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.

Cited by

• Laurent Sequences, Extended Rota Algebras and Categorical Discretization of Dynamical Systems
• A new discretization of the Euler equation via the finite operator theory

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• Quantum mechanics and umbral calculus
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