Factorization method applied to the second order q-difference operators
Introduction
In this paper we deal with a factorization method applied to the q-difference operators and associated eigenproblemfor . The basic idea of the factorization method is well known, see [5], [6], [7], [8]. One says that operators admit factorization if there exist sequences of first order q-difference operators and constants such thatIf the operators admit the factorization given by (2) then the eigenproblem (1) can be written in one of the following formsFrom these formulas we conclude, given that is an eigenvector of , then is a eigenvector for the operator and is a eigenvector for the operator . There exist some important classes of factorization with the property . In these cases any nonzero solution of equationis automatically a solution of Eq. (1). Moreover (3), (4) can be used to construct a solution of the eigenvalue problem for the operator .
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 .
Section snippets
Chain of factorized second order q-difference operators
Let us recall (see [4]) that the q-derivative and shift operators are defined byfor . The q-integral of the complex valued function defined on the q-intervalis given by
We are interested in the factorization of the sequence of the q-difference operatorsacting in the Hilbert space equipped with the scalar product
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 the operator tends to , the set becomes the interval and the scalar product turns to be . In the limiting case the operators are of the general form
- (a)
In the case when , then
Acknowledgment
The authors would like to thank T. Goliński for the careful reading of the manuscript and interest in this paper.
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