Elsevier

Applied Mathematics and Computation

Volume 228, 1 February 2014, Pages 147-152
Applied Mathematics and Computation

Factorization method applied to the second order q-difference operators

https://doi.org/10.1016/j.amc.2013.11.102Get rights and content

Abstract

We present certain classes of second order q-difference operators, which admit factorization into first order operators acting in a fixed Hilbert space. We also discuss classical limit case by letting q1.

Introduction

In this paper we deal with a factorization method applied to the q-difference operators and associated eigenproblemHkψk(x)Zk(x)qQ-1q+Wk(x)q+Vk(x)ψk(x)=λkψk(x)for kN{0}. The basic idea of the factorization method is well known, see [5], [6], [7], [8]. One says that operators Hk admit factorization if there exist sequences of first order q-difference operators Ak,Ak and constants ak,dk such thatHk=Ak*Ak+ak=dk+1-1Ak+1Ak+1+ak+1forkN{0}.If the operators Hk admit the factorization given by (2) then the eigenproblem (1) can be written in one of the following formsAkAkψk=(λk-ak)ψk,Ak+1Ak+1ψk=(dk+1λk-ak+1)ψk.From these formulas we conclude, given that ψk is an eigenvector of Hk, then Ak+1ψk is a eigenvector for the operator Hk+1 and Akψk is a eigenvector for the operator Hk-1. There exist some important classes of factorization with the property λ0=a0. In these cases any nonzero solution of equationA0ψ0=0is automatically a solution of Eq. (1). Moreover (3), (4) can be used to construct a solution ψk=AkA1ψ0 of the eigenvalue problem for the operator Hk.

In this work we will consider the classes of second order q-difference operators with factorization into first order operators acting in a fixed Hilbert space. We will present solutions of this problem and we will discuss the classical limit q1.

Section snippets

Chain of factorized second order q-difference operators

Let us recall (see [4]) that the q-derivative and shift operators are defined byqψ(x)ψ(x)-ψ(qx)(1-q)x,Qψ(x)ψ(qx)for 0<q<1. The q-integral of the complex valued function ψ:[a,b]qC defined on the q-interval[a,b]q{qna:nN{0}}{qnb:nN{0}}is given byabψ(x)dqxn=0(1-q)qnbψ(qnb)-aψ(qna).

We are interested in the factorization of the sequence of the q-difference operatorsHk=Zk(x)qQ-1q+Wk(x)q+Vk(x),kN{0}acting in the Hilbert space H=L2[a,b]q,ϱ(x)dqx equipped with the scalar productψ|φa

Limit case

Let us present the limit behavior of the formulas obtained above when the parameter q tends to 1. It is easy to see that in the limit case as q1 the operator q tends to ddx, the set [a,b]q becomes the interval [a,b] and the scalar product turns to be ψ|φ1abψ(x)φ(x)ϱ1(x)dx. In the limiting case the operators are of the general formAk1=ddx+fk1,Ak1=-ddx+fk1-(ϱ1)ϱ1,Hk1=-d2dx2-(ϱ1)ϱ1ddx-(fk1)+(fk1)2-fk1(ϱ1)ϱ1+ak1.

  • (a)

    In the case when r=-1, thenϱ1(x)=ϱ0xpe(b0-b1)x-ã22x2,fk1(x)=p2x+b0-b12+a0-a

Acknowledgment

The authors would like to thank T. Goliński for the careful reading of the manuscript and interest in this paper.

References (8)

  • B. Mielnik et al.

    The finite difference algorithm for higher order supersymmetry

    Phys. Lett. A

    (2000)
  • N.M. Atakishiyev et al.

    Explicit realization of the q-harmonic oscillator

    Theor. Math. Phys.

    (1991)
  • A. Dobrogowska et al.

    Second order q-difference equations solvable by factorization method

    J. Comput. Appl. Math.

    (2006)
  • A. Dobrogowska et al.

    Solutions of the q-deformed Schrödinger equation for special potentials

    J. Phys. A: Math. Theor.

    (2007)
There are more references available in the full text version of this article.

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