Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation

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.

41 pages, no figures

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Cited by in corpus (15)

• Continuous Symmetries of Difference Equations
• Group entropies, correlation laws and zeta functions
• Discretization of partial differential equations preserving their physical symmetries
• Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials
• (Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries
• On umbral extensions of Stirling numbers and Dobinski-like formulas
• Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
• Integrable maps from Galois differential algebras, Borel transforms and number sequences
• Umbral Deformations on Discrete Spacetime
• A superintegrable discrete oscillator and two-variable Meixner polynomials
• Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)
• Umbral Vade Mecum
• Equivalence and Symmetries for Linear Parabolic Equations and Applications Revisited
• The Jacobi last multiplier for difference equations
• Transmutations from the Covariant Transform on the Heisenberg Group and an Extended Umbral Principle