# Transformations of Difference Equations II

- Sonja Currie
^{1}Email author and - Anne D. Love
^{1}

**2010**:623508

**DOI: **10.1155/2010/623508

© S. Currie and A. D. Love. 2010

**Received: **13 April 2010

**Accepted: **6 September 2010

**Published: **14 September 2010

## Abstract

This is an extension of the work done by Currie and Love (2010) where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with non-eigenparameter-dependent boundary conditions at the end points. In particular, we now consider boundary conditions which depend affinely on the eigenparameter together with various combinations of Dirichlet and non-Dirichlet boundary conditions. The spectra of the resulting transformed boundary value problems are then compared to the spectra of the original boundary value problems.

## 1. Introduction

with representing a weight function and a potential function.

This paper is structured as follows.

The relevant results from [1], which will be used throughout the remainder of this paper, are briefly recapped in Section 2.

In Section 3, we show how non-Dirichlet boundary conditions transform to affine -dependent boundary conditions. In addition, we provide conditions which ensure that the linear function (in ) in the affine -dependent boundary conditions is a Nevanlinna or Herglotz function.

Section 4 gives a comparison of the spectra of all possible combinations of Dirichlet and non-Dirichlet boundary value problems with their transformed counterparts. It is shown that transforming the boundary value problem given by (2.2) with any one of the four combinations of Dirichlet and non-Dirichlet boundary conditions at the end points using (3.1) results in a boundary value problem with one extra eigenvalue in each case. This is done by considering the degree of the characteristic polynomial for each boundary value problem.

It is shown, in Section 5, that we can transform affine -dependent boundary conditions back to non-Dirichlet type boundary conditions. In particular, we can transform back to the original boundary value problem.

To conclude, we outline briefly how the process given in the sections above can be reversed.

## 2. Preliminaries

where and are constants, see [2]. Without loss of generality, by a shift of the spectrum, we may assume that the least eigenvalue, , of (1.1), (2.1) is .

Here, we take , thus is defined for .

## 3. Non-Dirichlet to Affine

to give obeying boundary conditions which depend affinely on the eigenparamter . We provide constraints which ensure that the form of these affine -dependent boundary conditions is a Nevanlinna/Herglotz function.

Theorem 3.1.

where , and . Here, and is a solution of (2.2) for , where is less than the least eigenvalue of (2.2), (3.2), and (3.13) such that for .

Proof.

The values of for which exists are . So to impose a boundary condition at , we need to extend the domain of to include . We do this by forcing the boundary condition (3.3) and must now show that the equation is satisfied on the extended domain.

where , and recall .

Note that for , this corresponds to the results in [1] for .

Theorem 3.2.

where , , and . Here, is a solution to (2.2) for , where is less than the least eigenvalue of (2.2), (3.2), and (3.13) such that in the given interval, .

Proof.

which is of the form (3.14), where , , and .

Note that if we require that in (3.3) be a Nevanlinna or Herglotz function, then we must have that . This condition provides constraints on the allowable values of .

Remark 3.3.

In Theorems 3.1 and 3.2, we have taken to be a solution of (2.2) for with less than the least eigenvalue of (2.2), (3.2), and (3.13) such that in . We assume that does not obey the boundary conditions (3.2) and (3.13) which is sufficient for the results which we wish to obtain in this paper. However, this case will be dealt with in detail in a subsequent paper.

Theorem 3.4.

Proof.

Since If , then we must have that either and or and . The first case of is not possible since and , , which implies that in particular for . For the second case, we get and which is not possible. Thus for , there are no allowable values of .

Also, if we require that from (3.14) be a Nevanlinna/Herglotz function, then we must have . This provides conditions on the allowable values of .

Corollary 3.5.

Proof.

Without loss of generality, we may shift the spectrum of (2.2) with boundary conditions (3.2), (3.13), such that the least eigenvalue of (2.2) with boundary conditions (3.2), (3.13) is strictly greater than , and thus we may assume that .

Since , we consider the two cases, and .

Now if , then the numerator of is strictly negative. Thus, in order that , we require that the denominator is strictly negative, that is, . So either and or and . As , we obtain that either and or and . These are the same conditions as we had on for . Thus, the sign of does not play a role in finding the allowable values of which ensure that , and hence we have the required result.

## 4. Comparison of the Spectra

In this section, we see how the transformation, (3.1), affects the spectrum of the difference equation with various boundary conditions imposed at the initial and terminal points.

By combining the results of [1, conclusion] with Theorems 3.1 and 3.2, we have proved the following result.

Theorem 4.1.

The transformation (3.1), where is a solution to (2.2) for , where is less than the least eigenvalue of (2.2) with one of the four sets of boundary conditions above, such that in the given interval , takes obeying (2.2) to obeying (2.5).

- (i)obeying (4.1) transforms to obeying(4.5)where and(4.6)
where with .

- (ii)
obeying (4.2) transforms to obeying (4.5) and (3.14).

- (iii)
obeying(4.3) transforms to obeying (3.3) and (4.6).

- (iv)
obeying (4.4) transforms to obeying (3.3) and (3.14).

The next theorem, shows that the boundary value problem given by obeying (2.2) together with any one of the four types of boundary conditions in the above theorem has eigenvalues as a result of the eigencondition being the solution of an th order polynomial in . It should be noted that if the boundary value problem considered is self-adjoint, then the eigenvalues are real, otherwise the complex eigenvalues will occur as conjugate pairs.

Theorem 4.2.

The boundary value problem given by obeying (2.2) together with any one of the four types of boundary conditions given by (4.1) to (4.4) has eigenvalues.

Proof.

where and are real constants, that is, a first order polynomial in .

where again are real constants, that is, a quadratic polynomial in .

where , and , are real constants, that is, an th and an th order polynomial in , respectively.

which is an th order polynomial in and, therefore, has roots. Hence, the boundary value problem given by obeying (2.2) with (4.1) has eigenvalues.

This is again an th order polynomial in and therefore has roots. Hence, the boundary value problem given by obeying (2.2) with (4.2) has eigenvalues.

where and are real constants, that is, a first order polynomial in .

where again , are real constants, that is, a quadratic polynomial in .

where , and , are real constants, thereby giving an th and an th order polynomial in , respectively.

which is an th order polynomial in and, therefore, has roots. Hence, the boundary value problem given by obeying (2.2) with (4.3) has eigenvalues.

This is again an th order polynomial in and therefore has roots. Hence, the boundary value problem given by obeying (2.2) with (4.4) has eigenvalues.

In a similar manner, we now prove that the transformed boundary value problems given in Theorem 4.1 have eigenvalues, that is, the spectrum increases by one in each case.

Theorem 4.3.

The boundary value problem given by obeying (2.5), , together with any one of the four types of transformed boundary conditions given in (i) to (iv) in Theorem 4.1 has eigenvalues. The additional eigenvalue is precisely with corresponding eigenfunction , as given in Theorem 4.1.

Proof.

The proof is along the same lines as that of Theorem 4.2. By Theorem 3.1, we have extended , such that exists for .

where , , , , and , are real constants, that is, an th, th, and th order polynomial in , respectively.

which is an th order polynomial in and thus has roots. Hence, the boundary value problem given by obeying (2.5) with transformed boundary conditions (i), that is, (4.5) and (4.6), has eigenvalues.

which is an th order polynomial in and thus has roots. Hence, the boundary value problem given by obeying (2.5) with transformed boundary conditions (ii), that is, (4.5) and (3.14), has eigenvalues.

where , , and are real constants.

where , , , , and are real constants.

where all the coefficients of are real constants.

which is an th order polynomial in and thus has roots. Hence, the boundary value problem given by obeying (2.5) with transformed boundary conditions (iii), that is, (3.3) and (4.6), has eigenvalues.

which is an th order polynomial in and thus has roots. Hence, the boundary value problem given by obeying (2.5) with transformed boundary conditions (iv), that is, (3.3) and (3.14), has eigenvalues.

Lastly, we have that (3.1) transforms eigenfunctions of any of the boundary value problems in Theorem 4.2 to eigenfunctions of the corresponding transformed boundary value problem, see Theorem 4.2. In particular, if are the eigenvalues of the original boundary value problem with corresponding eigenfunctions , then are eigenfunctions of the corresponding transformed boundary value problem with eigenvalues . Since we know that the transformed boundary value problem has eigenvalues, it follows that constitute all the eigenvalues of the transformed boundary value problem, see [1].

## 5. Affine to Non-Dirichlet

we can transform obeying affine -dependent boundary conditions to obeying non-Dirichlet boundary conditions.

Theorem 5.1.

where and .

Proof.

So to impose the boundary condition (5.7), it is necessary to extend the domain of by forcing the boundary condition (5.7). We must then check that satisfies the equation on the extended domain.

Note that the case of , that is, a non-Dirichlet boundary condition, gives , that is, which corresponds to the results obtained in [1].

Remark 5.2.

That is, the boundary value problem given by satisfying (2.5) with boundary conditions (5.2), (5.3) transforms under (5.1) to obeying (2.2) with boundary conditions (3.2), (3.13) which is the original boundary value problem.

hence .

Thus, , that is, .

that is, .

To summarise, we have the following.

- (a)
non-Dirichlet and non-Dirichlet, that is, (4.5) and (4.6);

- (b)
non-Dirichlet and affine, that is, (4.5) and (3.14);

- (c)
affine and non-Dirichlet, that is, (3.3) and (4.6);

- (d)
affine and affine, that is, (3.3) and (3.14).

By Theorem 4.3, each of the above boundary value problems have eigenvalues.

- (1)
boundary conditions (a) transform to and ;

- (2)
boundary conditions (b) transform to and (3.13);

- (3)
boundary conditions (c) transform to (3.2) and ;

- (4)
boundary conditions (d) transform to (3.2) and (3.13).

By Theorem 4.2, we know that the above transformed boundary value problems in each have eigenvalues. In particular, if are the eigenvalues of (2.5), (a) ((b), (c), (d), resp.) with eigenfunctions , then and are eigenfunctions of (2.2), (1) ((2), (3), (4), resp.) with eigenvalues . Since we know that these boundary value problems have eigenvalues, it follows that constitute all the eigenvalues.

## 6. Conclusion

- (i)
non-Dirichlet at the initial point and affine at the terminal point;

- (ii)
affine at the initial point and non-Dirichlet at the terminal point;

- (iii)
affine at the initial point and at the terminal point.

- (A)
Dirichlet at the initial point and non-Dirichlet at the terminal point;

- (B)
non-Dirichlet at the initial point and Dirichlet at the terminal point;

- (C)
non-Dirichlet at the initial point and at the terminal point.

It is then possible to return to the original boundary value problem by applying a suitable transformation to the transformed boundary value problem above.

## Declarations

### Acknowledgments

The authors would like to thank Professor Bruce A. Watson for his useful input and suggestions. This work was supported by NRF Grant nos. TTK2007040500005 and FA2007041200006.

## Authors’ Affiliations

## References

- Currie S, Love A:
**Transformations of difference equations I.***Advances in Difference Equations*2010,**2010:**-22.Google Scholar - Atkinson FV:
*Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, vol. 8*. Academic Press, New York, NY, USA; 1964:xiv+570.Google Scholar

## Copyright

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