# Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice

@article{Dimakis1995UmbralCD, title={Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice}, author={Aristophanes Dimakis and Folkert Mueller-Hoissen and T. Striker}, journal={Journal of Physics A}, year={1995}, volume={29}, pages={6861-6876} }

`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the… Expand

#### 31 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

Umbral calculus, difference equations and the discrete Schrödinger equation

- Mathematics, Physics
- 2004

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… Expand

Lattice gauge fields and noncommutative geometry

- Mathematics, Physics
- 1998

Abstract Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this… Expand

Integrable maps from Galois differential algebras, Borel transforms and number sequences

- Mathematics, Physics
- 2013

Abstract A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves… Expand

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2014

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature ( 0, n ) the umbral calculus framework with Lie-al algebraic symmetries with certain applications from the view of interpolation theory and integral transforms are discussed. Expand

(DISCRETE) ALMANSI TYPE DECOMPOSITIONS: AN UMBRAL CALCULUS FRAMEWORK BASED ON osp(1|2) SYMMETRIES

- Mathematics, Physics
- 2011

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multi- variate polynomials IRŒxshall be described in terms of the generators of the… 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

Lorentz and Galilei Invariance on Lattices

- Physics
- 2004

We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical… Expand

From symmetries to number theory

- Physics
- 2009

It is shown that the finite-operator calculus provides a simple formalism useful for constructing symmetry-preserving discretizations of quantum-mechanical integrable models. A related algebraic… 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

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