Umbral calculus, difference equations and the discrete Schrödinger 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.

Discrete Integrable Systems

“…A different approach was developed mainly for linear or linearizable difference equations and involved transformations acting on more than one point of the lattice [20,21,39,28,36]. The symmetries considered in this approach are really generalized ones, however they reduce to point ones in the continuous limit.…”

“…For a proof see [36]. Let us assume that we know the solution of an umbral equation for ∆ = ∂ x and it has the form…”

Heisenberg algebra, umbral calculus and orthogonal polynomials

“…A recent article [12] was devoted to a systematic study of the realization ofP as a difference operator andM as an operator the projection of which is the coordinate x. This was applied to show that continuous symmetries like Lorentz or Galilei invariance can be implemented in quantum theories on lattices [12,13].…”

Umbral orthogonal polynomials

“…The umbral calculus [1,2,3] is a powerful mathematical tool allowing us to discretize in a systematic way linear differential equations as well as their solutions preserving their underlying algebraic structures, in particular, symmetry groups and integrability properties [4,5]. The umbral discretization of a linear differential equation gives a difference equation.…”

“…Some attempts about discretization of quantum mechanics has been made either using umbral calculus for the discretization of the Schrödinger equation [5,10,11,12] or via its reformulation in a discrete space-time (see Ref. [13] and references therein).…”