Lie-algebraic discretization of differential equations

TL;DR

This paper introduces a Lie-algebraic method for discretizing differential equations that preserves spectral properties and reproduces classical orthogonal polynomials in a finite-difference framework.

Contribution

It develops a novel Lie-algebraic discretization approach using $sl_2$-algebra, enabling the construction of (quasi)-exact solutions and reproducing classical polynomials.

Findings

Reproduces Hahn, Charlier, and Meixner polynomials as eigenfunctions.

Introduces a discrete version of Hermite, Laguerre, Legendre, and Jacobi polynomials.

Provides a finite-difference discretization preserving spectral properties.

Abstract

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 is proposed. Using $sl_2$-algebra based approach, (quasi)-exactly-solvable finite-difference equations are described. It is shown that the operators having the Hahn, Charlier and Meixner polynomials as the eigenfunctions are reproduced in present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.