Lie - algebraic discretization of differential equations by Yuri Smirnov

A certain representation for the Heisenberg algebra in finitedifference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl2-algebra based approach, (quasi)-exactlysolvable 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. On leave of absence from the Nuclear Physics Institute, Moscow State University, Moscow 119899, Russia E-mail: [email protected] On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia E-mail: [email protected] or [email protected] Discretization is one of the most powerful tools of solving continuous theories. It appears in different forms in various physics sciences leading to discrete versions of differential equations of classical mechanics, lattice field theories etc. However, once the discretization is chosen as an approach to a problem we meet a hard problem of ambiguity – there exist infinitely-many different ways of discretization having the same continuous limit. The goal of present paper is to demonstrate a certain discretization scheme for differential equations with exceptional properties – eigenvalues remain unchanged (isospectrality), eigenfunctions are modified in a simple manner etc. 1. Two operators a and b obeying the commutation relation [a, b] ≡ ab− ba = 1, (1) with the identity operator in the r.h.s. define the Heisenberg algebra. The Heisenberg algebra plays the central role in many branches of theoretical and mathematical physics. A standard representation of (1) in the action on the real line is the coordinate-momentum one: a = d dx , b = x . (2) Our goal is to build the one-parametric representation of (1) on the line in terms of finite-difference operators having (2) as a limiting case. Let us introduce the finite-difference operatorsD± possessing the property of the translation covariance D+f(x) = f(x+ δ)− f(x) δ ≡ (e d dx − 1) δ f(x) (3) and D−f(x) = f(x)− f(x− δ) δ ≡ (1− e d dx ) δ f(x) (4) and D+ → D−, once δ → −δ. It is worth noting that D+ −D− = δD−D+ (5) Now define the so-called quasi-monomial x = x(x− δ)(x− 2δ) . . . (x− nδ) = δ Γ( δ + 1) Γ( δ − n) , (6)