Transformations of Difference Equations I
© Sonja Currie and Anne D. Love. 2010
Received: 13 April 2010
Accepted: 29 July 2010
Published: 10 August 2010
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.
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 [3–5] to study the transformations of difference equations.
can be factorised as , however, reversing the factors that is, finding 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  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 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 of (1.1), with 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 , 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 particles at the points say with masses and distances between them given by , . Beyond the string extends to a length and beyond to a length . The string is stretched to unit tension. If is the displacement of the particle at time , the restoring forces on it due to the tension of the string are and 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 , where is the amplitude of oscillation of the particle . Solving for 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  for details. Secondly, there is an equivalent scenario in electrical network theory. In this case, the are inductances, capacitances, and the 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 . 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 , 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 th new norming constant is just multiplied by the original th norming constant or multiplied by the original 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, , to (1.1) for with 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 .
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 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
which is the required transformed equation.
Thus we obtain (2.5).
2.2. Transformation of the Boundary Conditions
We call (2.14) the transformed boundary conditions.
Combining the above results we obtain the following corollary.
The transformation , 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, , which has been removed.
Theorems 2.1 and 2.2 prove that the mapping 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 eigenvalues which are real and distinct and the corresponding eigenfunctions are linearly independent when considered for ; see  for the case of vector difference equations of which the above is a special case. In particular, if are the eigenvalues of (1.1), (2.2) with eigenfunctions , then and are eigenfunctions of (2.5), (2.14) with eigenvalues . By a simple computation it can be shown that . 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 constitute all the eigenvalues of (2.5), (2.14).
2.3. Transformation of the Norming Constants
Let be the eigenvalues of (1.1) with boundary conditions (2.2) and be associated eigenfunctions normalised by . We prove, in this subsection, that under the mapping given in Theorem 2.2, the new norming constant is times the original norming constant.
3. Transformation 2
3.1. Transformation of the Equation
3.2. Transformation of the Boundary Conditions
The mapping given by , , where is as previously defined (in the beginning of the section), transforms which obeys boundary conditions (2.14) to which obeys the non-Dirichlet boundary conditions (3.9) and is a solution of for .
Combining Theorems 3.1 and 3.2 we obtain the corollary below.
Let be a solution of (2.5) for , where is less than the least eigenvalue of (2.5), (2.14), such that for . 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, .
Theorems 3.1 and 3.2 prove that the mapping , transforms eigenfunctions of (2.5), (2.14) to eigenfunctions of (3.3), (3.9). In particular if are the eigenvalues of (2.5), (2.14), , with eigenfunctions , then are eigenfunctions of (3.3), (3.9), , with eigenvalues . 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 constitute all the eigenvalues of (3.3), (3.9).
Thus we have proved the following.
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).
which, when we substitute in for , , and , simplifies to zero. Obviously the right-hand side of (2.5) is equal to for . Thus is a solution of (2.5) for , where is less than the least eigenvalue of (2.5), (2.14).
Hence, as claimed, we have proved the following result.
3.3. Transformation of the Norming Constants
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.
can be extended to include and by forcing (3.9). Here is a solution of (2.5) for , with less than the least eigenvalue of (2.5), (2.14) such that for all . The mapping then gives that 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, .
Let , where is a solution of (2.5) with , where is less than the least eigenvalue of (2.5), (2.14), such that for all . Then is an eigenfunction of (3.3), (3.9) corresponding to the eigenvalue , where we define via and .
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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