Factorization Method and General Second Order Linear Difference Equation

Abstract

This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually adjoint first order difference operators. These classes encompass equations of hypergeometic type describing classical orthogonal polynomials of a discrete variable.

Discrete Quantum Harmonic Oscillator

“…This type of factorizations was presented in detail for the general situation in a discrete case in [16][17][18] and in [19][20][21] for the τ-and q-cases too. It is based on classical methods taken from the work of some founders of quantum mechanics such as Schrödinger [22] (see also [23][24][25]).…”

“…In general, the creation operator A * k is the adjoint of the annihilation operator A k relative to the scalar product (6), where the weight functions ρ k satisfy the Pearson type equations (see [17]). In our case, we have…”

Darboux transformations and second order difference equations

“…This article is an extension of previous works concerning the factorization method applied to second order differential and difference operators [32,23,26,29,1,2,10,11,12,22,31]. Some results obtained in [8,9,27] are used and adapted to our context. As general references related to these subjects, we recommend [3,13,17,18,19,28,30].…”