Umbral calculus, difference equations and the discrete Schrödinger equation

@article{Levi2004UmbralCD,
  title={Umbral calculus, difference equations and the discrete Schr{\"o}dinger equation},
  author={D. Levi and P. Tempesta and P. Winternitz},
  journal={Journal of Mathematical Physics},
  year={2004},
  volume={45},
  pages={4077-4105}
}
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 Schrodinger 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… Expand
Quantum mechanics and umbral calculus
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 theExpand
Discrete q-derivatives and symmetries of q-difference equations
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariantExpand
Heisenberg algebra, umbral calculus and orthogonal polynomials
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shallExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Heat Polynomials, Umbral Correspondence and Burgers Equations
We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the generalExpand
The discrete and periodic heat and harmonic oscillator equation
In this paper we present a generalization of the famous Dirac ladder formalism for the Schrödinger harmonic oscillator equation. Whereas the classical one-dimensional harmonic oscillator acts onExpand
Discretization of partial differential equations preserving their physical symmetries
A procedure for obtaining a 'minimal' discretization of a partial differential equation, preserving all of its Lie point symmetries, is presented. 'Minimal' in this case means that the differentialExpand
Discretization of superintegrable systems on a plane
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 generalizedExpand
Multiple-scale analysis of dynamical systems on the lattice
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 dynamicalExpand
Discretization of nonlinear evolution equations over associative function algebras
Abstract A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way thatExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 78 REFERENCES
Discrete q-derivatives and symmetries of q-difference equations
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariantExpand
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`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 andExpand
Lie-algebraic discretization of differential equations
A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property isExpand
Lie symmetries of multidimensional difference equations
A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. TheyExpand
Continuous symmetries of Lagrangians and exact solutions of discrete equations
One of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of theExpand
Symmetries and integrability of difference equations
The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinearExpand
Discrete derivatives and symmetries of difference equations
We show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable for finding the symmetries of discrete equations. InExpand
Lie algebra contractions and symmetries of the Toda hierarchy
The Lie algebra L(Δ) of generalized and point symmetries of the equations in the Toda hierarchy is shown to be a semidirect sum of two infinite-dimensional Lie algebras, one perfect, the otherExpand
Lie symmetries of finite‐difference equations
Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations areExpand
Symmetries of the discrete Burgers equation
A discrete Cole-Hopf transformation is used to derive a discrete Burgers equation that is linearizable to a discrete heat equation. A five-dimensional symmetry algebra is obtained that reduces to theExpand
...
1
2
3
4
5
...