Advances in Difference Equations

, 2010:947058 | Cite as

Transformations of Difference Equations I

Open Access
Research Article

Abstract

We consider a general weighted second-order difference equation. Two transformations are studied which transform the given equation into another weighted second order difference equation of the same type, these are based on the Crum transformation. We also show how Dirichlet and non-Dirichlet boundary conditions transform as well as how the spectra and norming constants are affected.

Keywords

Difference Equation Dirichlet Boundary Condition Norming Constant Jacobi Operator Original Boundary 

1. Introduction

Our interest in this topic arose from the work done on transformations and factorisations of continuous (as opposed to discrete) Sturm-Liouville boundary value problems by, amongst others, Binding et al., notably [1, 2]. We make use of similar ideas to those discussed in [3, 4, 5] to study the transformations of difference equations.

In this paper, we consider a weighted second-order difference equation of the form

where Open image in new window represents a weight function and Open image in new window a potential function.

Two factorisations of the formal difference operator, Open image in new window , associated with (1.1), are given. Although there may be many alternative factorisations of this operator (see e.g., [2, 6]), the factorisations given in Theorems 2.1 and 3.1 are of particular interest to us as they are analogous to those used in the continuous Sturm-Liouville case. Moreover, if the original operator is factorised by Open image in new window , as in Theorem 2.1, or by Open image in new window , as in Theorem 3.1, then the Darboux-Crum type transformation that we wish to study is given by the mapping Open image in new window or Open image in new window , respectively. This results in eigenfunctions of the difference boundary value problem being transformed to eigenfunctions of another, so-called, transformed boundary value problem given by permuting the factors Open image in new window and Open image in new window or the factors Open image in new window and Open image in new window , that is, by Open image in new window or Open image in new window , respectively, as in the continuous case. Applying this transformation must then result in a transformed equation of exactly the same type as the original equation. In order to ensure this, we require that the original difference equation which we consider has the form given in (1.1). In particular the weight, Open image in new window , also determines the dependence on the off-diagonal elements. We note that the more general equation

can be factorised as Open image in new window , however, reversing the factors that is, finding Open image in new window does not necessarily result in a transformed equation of the same type as (1.2). The importance of obtaining a transformed equation of exactly the same form as the original equation, is that ultimately we will (in a sequel to the current paper) use these transformations to establish a hierarchy of boundary value problems with (1.1) and various boundary conditions; see [4] for the differential equations case. Initially we transform, in this paper, non-Dirichlet boundary conditions to Dirichlet boundary conditions and back again. In the sequel to this paper, amongst other things, non-Dirichlet boundary conditions are transformed to boundary conditions which depend affinely on the eigenparameter Open image in new window and vice versa. At all times, it is possible to keep track of how the eigenvalues of the various transformed boundary value problems relate to the eigenvalues of the original boundary value problem.

The transformations given in Theorems 2.1 and 3.1 are almost isospectral. In particular, depending on which transformation is applied at a specific point in the hierarchy, we either lose the least eigenvalue or gain an eigenvalue below the least eigenvalue. It should be noted that if we apply the two transformations of Sections 2 and 3 successively the resulting boundary value problem has precisely the same spectrum as the boundary value problem we began with. In fact, for a suitable choice of the solution Open image in new window of (1.1), with Open image in new window less than the least eigenvalue of the boundary value problem fixed, Corollary 3.3 gives that applying the transformation given in Theorem 2.1 followed by the transformation given in Theorem 3.1 yields a boundary value problem which is exactly the same as the original boundary value problem, that is, the same difference equation, boundary conditions, and hence spectrum.

It should be noted that the work [6, Chapter 11] of Teschl, on spectral and inverse spectral theory of Jacobi operators, provides a factorisation of a second-order difference equation, where the factors are adjoints of each another. It is easy to show that the factors given in this paper are not adjoints of each other, making our work distinct from that of Teschl's.

Difference equations, difference operators, and results concerning the existence and construction of their solutions have been discussed in [7, 8]. Difference equations occur in a variety of settings, especially where there are recursive computations. As such they have applications in electrical circuit analysis, dynamical systems, statistics, and many other fields.

More specifically, from Atkinson [9], we obtained the following three physical applications of the difference equation (1.1). Firstly, we have the vibrating string. The string is taken to be weightless and bears Open image in new window particles Open image in new window at the points say Open image in new window with masses Open image in new window and distances between them given by Open image in new window , Open image in new window . Beyond Open image in new window the string extends to a length Open image in new window and beyond Open image in new window to a length Open image in new window . The string is stretched to unit tension. If Open image in new window is the displacement of the particle Open image in new window at time Open image in new window , the restoring forces on it due to the tension of the string are Open image in new window and Open image in new window considering small oscillations only. Hence, we can find the second-order differential equation of motion for the particles. We require solutions to be of the form Open image in new window , where Open image in new window is the amplitude of oscillation of the particle Open image in new window . Solving for Open image in new window then reduces to solving a difference equation of the form (1.1). Imposing various boundary conditions forces the string to be pinned down at one end, both ends, or at a particular particle, see Atkinson [9] for details. Secondly, there is an equivalent scenario in electrical network theory. In this case, the Open image in new window are inductances, Open image in new window capacitances, and the Open image in new window are loop currents in successive meshes. The third application of the three-term difference equation (1.1) is in Markov processes, in particular, birth and death processes and random walks. Although the above three applications are somewhat restricted due to the imposed relationship between the weight and the off-diagonal elements, they are nonetheless interesting.

There is also an obvious connection between the three-term difference equation and orthogonal polynomials; see [10]. Although, not the focus of this paper, one can investigate which orthogonal polynomials satisfy the three-term recurrence relation given by (1.1) and establish the properties of those polynomials. In Atkinson [9], the link between the norming constants and the orthogonality of polynomials obeying a three-term recurrence relation is given. Hence the necessity for showing how the norming constants are transformed under the transformations given in Theorems 2.1 and 3.1. As expected, from the continuous case, we find that the Open image in new window th new norming constant is just Open image in new window multiplied by the original Open image in new window th norming constant or Open image in new window multiplied by the original Open image in new window th norming constant depending on which transformation is used.

The paper is set out as follows.

In Section 2, we transform (1.1) with non-Dirichlet boundary conditions at both ends to an equation of the same form but with Dirichlet boundary conditions at both ends. We prove that the spectrum of the new boundary value problem is the same as that of the original boundary value problem but with one eigenvalue less, namely, the least eigenvalue.

In Section 3, we again consider an equation of the form (1.1), but with Dirichlet boundary conditions at both ends. We assume that we have a strictly positive solution, Open image in new window , to (1.1) for Open image in new window with Open image in new window less than the least eigenvalue of the given boundary value problem. We can then transform the given boundary value problem to one consisting of an equation of the same type but with specified non-Dirichlet boundary conditions at the ends. The spectrum of the transformed boundary value problem has one extra eigenvalue, in particular Open image in new window .

The transformation in Section 2 followed by the transformation in Section 3, gives in general, an isospectral transformation of the weighted second-order difference equation of the form (1.1) with non-Dirichlet boundary conditions. However, for a particular choice of Open image in new window this results in the original boundary value problem being recovered.

In the final section, we show that the process outlined in Sections 2 and 3 can be reversed.

2. Transformation 1

2.1. Transformation of the Equation

Consider the second-order difference equation (1.1), which may be rewritten as
where Open image in new window . Denote by Open image in new window the least eigenvalue of (1.1) with boundary conditions
where Open image in new window and Open image in new window are constants; see [9]. We wish to find a factorisation of the formal operator,

for Open image in new window , such that Open image in new window , where Open image in new window and Open image in new window are both first order formal difference operators.

Theorem 2.1.

Let Open image in new window be a solution of (1.1) corresponding to Open image in new window and define the formal difference operators
Then formally Open image in new window , Open image in new window and the so-called transformed operator is given by Open image in new window , Open image in new window . Hence the transformed equation is

Proof.

By the definition of Open image in new window and Open image in new window , we have that
Using (2.3), substituting in for Open image in new window and cancelling terms, gives

Hence Open image in new window .

which is the required transformed equation.

To find Open image in new window , we need to determine Open image in new window .

Firstly,
By multiplying by Open image in new window , this may be rewritten as

Thus we obtain (2.5).

2.2. Transformation of the Boundary Conditions

We now show how the non-Dirichlet boundary conditions (2.2) are transformed under Open image in new window .

By the boundary conditions (2.2) Open image in new window is defined for Open image in new window .

Theorem 2.2.

The mapping Open image in new window given by Open image in new window , Open image in new window , where Open image in new window is an eigenfunction to the least eigenvalue Open image in new window of (1.1), (2.2), transforms Open image in new window obeying boundary conditions (2.2) to Open image in new window obeying Dirichlet boundary conditions of the form

Proof.

Hence as Open image in new window obeys the non-Dirichlet boundary condition Open image in new window , Open image in new window obeys the Dirichlet boundary condition, Open image in new window .

Similarly, for the second boundary condition,

We call (2.14) the transformed boundary conditions.

Combining the above results we obtain the following corollary.

Corollary 2.3.

The transformation Open image in new window , given in Theorem 2.2, takes eigenfunctions of the boundary value problem (1.1), (2.2) to eigenfunctions of the boundary value problem (2.5), (2.14). The spectrum of the transformed boundary value problem (2.5), (2.14) is the same as that of (1.1), (2.2), except for the least eigenvalue, Open image in new window , which has been removed.

Proof.

Theorems 2.1 and 2.2 prove that the mapping Open image in new window transforms eigenfunctions of (1.1), (2.2) to eigenfunctions (or possibly the zero solution) of (2.5), (2.14). The boundary value problem (1.1), (2.2) has Open image in new window eigenvalues which are real and distinct and the corresponding eigenfunctions Open image in new window are linearly independent when considered for Open image in new window ; see [11] for the case of vector difference equations of which the above is a special case. In particular, if Open image in new window are the eigenvalues of (1.1), (2.2) with eigenfunctions Open image in new window , then Open image in new window and Open image in new window are eigenfunctions of (2.5), (2.14) with eigenvalues Open image in new window . By a simple computation it can be shown that Open image in new window . Since the interval of the transformed boundary value problem is precisely one shorter than the original interval, (2.5), (2.14) has one less eigenvalue. Hence Open image in new window constitute all the eigenvalues of (2.5), (2.14).

2.3. Transformation of the Norming Constants

Let Open image in new window be the eigenvalues of (1.1) with boundary conditions (2.2) and Open image in new window be associated eigenfunctions normalised by Open image in new window . We prove, in this subsection, that under the mapping given in Theorem 2.2, the new norming constant is Open image in new window times the original norming constant.

Lemma 2.4.

Let Open image in new window denote the norming constants of (1.1) and be defined by

Proof.

Then, using the definition of Open image in new window , we obtain that
Therefore,
Using (1.1) to substitute in for Open image in new window and Open image in new window gives

Theorem 2.5.

If Open image in new window , as defined in Lemma 2.4, are the norming constants of (1.1) with boundary conditions (2.2) and
are the norming constants of (2.5) with boundary conditions (2.14), then

Proof.

The boundary conditions (2.2) together with Lemma 2.4 give
Now by (2.14), Open image in new window , and thus
Therefore,
Thus we have that

3. Transformation 2

3.1. Transformation of the Equation

Consider (2.5), where Open image in new window and Open image in new window , Open image in new window , obeys the boundary conditions (2.14).

Let Open image in new window be a solution of (2.5) with Open image in new window such that Open image in new window for all Open image in new window , where Open image in new window is less than the least eigenvalue of (2.5), (2.14).

We want to factorise the operator Open image in new window , where

for Open image in new window such that Open image in new window , where Open image in new window and Open image in new window are both formal first order difference operators.

Theorem 3.1.

Proof.

Hence Open image in new window .

giving that Open image in new window is a solution of the transformed equation.

We now explicitly obtain the transformed equation. From the definitions of Open image in new window and Open image in new window , we get
This implies that

3.2. Transformation of the Boundary Conditions

At present, Open image in new window is defined for Open image in new window . We extend the definition of Open image in new window to Open image in new window by forcing the boundary conditions

Here we take Open image in new window .

Theorem 3.2.

The mapping Open image in new window given by Open image in new window , Open image in new window , where Open image in new window is as previously defined (in the beginning of the section), transforms Open image in new window which obeys boundary conditions (2.14) to Open image in new window which obeys the non-Dirichlet boundary conditions (3.9) and Open image in new window is a solution of Open image in new window for Open image in new window .

Proof.

By the construction of Open image in new window and Open image in new window it follows that the boundary conditions (3.9) are obeyed by Open image in new window .

We now show that Open image in new window is a solution to the extended problem. From Theorem 3.1 we need only prove that Open image in new window for Open image in new window and Open image in new window . For Open image in new window , from (3.3) with (3.9), we have that
Also the mapping, for Open image in new window , gives
Thus using (2.14), we obtain that Open image in new window . So we now have
Next, using the mapping at Open image in new window , we obtain that
Rearranging the terms above results in
Also, (2.5), for Open image in new window , together with (2.14) gives
Subtracting (3.15) from (3.16) yields

In a similar manner, we can show that (3.3) also holds for Open image in new window . Hence Open image in new window is a solution of Open image in new window for Open image in new window .

Combining Theorems 3.1 and 3.2 we obtain the corollary below.

Corollary 3.3.

Let Open image in new window be a solution of (2.5) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.5), (2.14), such that Open image in new window for Open image in new window . Then we can transform the given equation, (2.5), to an equation of the same type, (3.3) with a specified non-Dirichlet boundary condition, (3.9), at either the initial or end point. The spectrum of the transformed boundary value problem (3.3), (3.9) is the same as that of (2.5), (2.14) except for one additional eigenvalue, namely, Open image in new window .

Proof.

Theorems 3.1 and 3.2 prove that the mapping Open image in new window , transforms eigenfunctions of (2.5), (2.14) to eigenfunctions of (3.3), (3.9). In particular if Open image in new window are the eigenvalues of (2.5), (2.14), Open image in new window , with eigenfunctions Open image in new window , then Open image in new window are eigenfunctions of (3.3), (3.9), Open image in new window , with eigenvalues Open image in new window . Since the index set of the transformed boundary value problem is precisely one larger than the original, (3.3), (3.9) has one more eigenvalue. Hence Open image in new window constitute all the eigenvalues of (3.3), (3.9).

Thus we have proved the following.

Corollary 3.4.

The transformation of (1.1), (2.2) to (2.5), (2.14) and then to (3.3), (3.9) is an isospectral transformation. That is, the spectrum of (1.1), (2.2) is the same as the spectrum of (3.3), (3.9).

We now show that for a suitable choice of Open image in new window the transformation of (1.1), (2.2) to (2.5), (2.14) and then to (3.3), (3.9) results in the original boundary value problem.

Without loss of generality, by a shift of the spectrum, it may be assumed that the least eigenvalue, Open image in new window , of (1.1), (2.2) is Open image in new window . Furthermore, let Open image in new window be an eigenfunction to (1.1), (2.2) for the eigenvalue Open image in new window .

Theorem 3.5.

If Open image in new window , then Open image in new window is a solution of (2.5), for Open image in new window . Here Open image in new window is less than the least eigenvalue of (2.5), (2.14) and Open image in new window has no zeros in the interval Open image in new window . In addition, Open image in new window , Open image in new window , Open image in new window for Open image in new window and Open image in new window for Open image in new window .

Proof.

The left hand-side of (2.5), with Open image in new window , becomes

which, when we substitute in for Open image in new window , Open image in new window , and Open image in new window , simplifies to zero. Obviously the right-hand side of (2.5) is equal to Open image in new window for Open image in new window . Thus Open image in new window is a solution of (2.5) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.5), (2.14).

Substituting for Open image in new window , Open image in new window and Open image in new window , in the equation for Open image in new window , we obtain immediately that Open image in new window for Open image in new window and by assumption Open image in new window .

Next, a similar substitution into the equation for Open image in new window yields
But Open image in new window is an eigenfunction of (1.1), (2.2) corresponding to the eigenvalue Open image in new window , thus
Lastly, by definition
Putting this into the equation for Open image in new window above yields
Substituting in for Open image in new window gives

so by (2.2) Open image in new window .

Using precisely the same method, it can be easily shown that Open image in new window .

Hence, as claimed, we have proved the following result.

Corollary 3.6.

The transformation of (1.1), (2.2) to (2.5), (2.14) and then to (3.3), (3.9) with Open image in new window results in the original boundary value problem.

3.3. Transformation of the Norming Constants

Assume that we have the following normalisation: Open image in new window . A result analogous to that in Theorem 2.5 is obtained.

Lemma 3.7.

As before let Open image in new window denote the norming constants of (2.5) and be defined by

Proof.

Substituting in for Open image in new window and Open image in new window , we obtain that
Then, using the definition of Open image in new window , we obtain that

Theorem 3.8.

If Open image in new window , as given in Lemma 3.7, are the norming constants of (2.5) with boundary conditions (2.14) and
are the norming constants of (3.3) with boundary conditions (3.9), then

Proof.

Using the boundary conditions (2.14) together with Lemma 3.7, we obtain that
Therefore,

4. Conclusion

To conclude, we illustrate how the process may be done the other way around. To do this we start by transforming a second-order difference equation with Dirichlet boundary conditions at both ends to a second-order difference equation of the same type with non-Dirichlet boundary conditions at both ends and then transform this back to the original boundary value problem.

Consider Open image in new window such that Open image in new window satisfies (2.5) and (2.14). The mapping Open image in new window , given by

can be extended to include Open image in new window and Open image in new window by forcing (3.9). Here Open image in new window is a solution of (2.5) for Open image in new window , with Open image in new window less than the least eigenvalue of (2.5), (2.14) such that Open image in new window for all Open image in new window . The mapping Open image in new window then gives that Open image in new window satisfies (3.3) and (3.9). So (3.3), (3.9) has the same spectrum as (2.5), (2.14) except that one eigenvalue has been added, namely, Open image in new window .

Now the mapping Open image in new window given by
where Open image in new window is an eigenfunction of (3.3), (3.9) corresponding to the eigenvalue Open image in new window yields
with boundary conditions

Thus this boundary value problem in Open image in new window has the same spectrum as that of (3.3), (3.9) but with one eigenvalue removed, namely, Open image in new window .

Lemma 4.1.

Let Open image in new window , where Open image in new window is a solution of (2.5) with Open image in new window , where Open image in new window is less than the least eigenvalue of (2.5), (2.14), such that Open image in new window for all Open image in new window . Then Open image in new window is an eigenfunction of (3.3), (3.9) corresponding to the eigenvalue Open image in new window , where we define Open image in new window via Open image in new window and Open image in new window .

Proof.

By construction, we have that
Hence substituting the expressions for Open image in new window , and Open image in new window into the above equation yields

Thus Open image in new window obeys the boundary conditions (3.9).

Next, we show that Open image in new window solves (3.3). Substituting in for Open image in new window , we obtain that, for Open image in new window ,
Using the expressions for Open image in new window and Open image in new window , we obtain by direct substitution that

Now, if we examine the right-hand side of (3.3), we immediately see that it is equal to Open image in new window for Open image in new window .

Substituting in for Open image in new window , and Open image in new window , we obtain that
Note that
which when we substitute in for Open image in new window and Open image in new window becomes
Therefore,
giving by a straightforward calculation

Thus Open image in new window is a solution for (3.3), Open image in new window and hence an eigenfunction of (3.3), (3.9) corresponding to the eigenvalue Open image in new window .

Theorem 4.2.

The boundary value problem (4.3) with boundary conditions

is the same as the original boundary value problem for Open image in new window , that is, (2.5) and (2.14), where Open image in new window is as in Lemma 4.1.

Proof.

All we need to show is that Open image in new window and Open image in new window . Substituting in for Open image in new window gives directly that
Using the expression for Open image in new window , we obtain that

Notes

Acknowledgments

The authors would like to thank Professor Bruce A. Watson for his ideas, guidance, and assistance. This paper was supported by NRF Grant nos. TTK2007040500005 and FA2007041200006.

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Copyright information

© Sonja Currie and Anne D. Love. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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