# Transformations of Difference Equations II

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## 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.

## Keywords

Boundary Condition Difference Equation Initial Point Dirichlet Boundary Condition Characteristic Polynomial## 1. Introduction

with Open image in new window representing a weight function and Open image in new window 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 Open image in new window -dependent boundary conditions. In addition, we provide conditions which ensure that the linear function (in Open image in new window ) in the affine Open image in new window -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 Open image in new window -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 Open image in new window and Open image in new window are constants, see [2]. Without loss of generality, by a shift of the spectrum, we may assume that the least eigenvalue, Open image in new window , of (1.1), (2.1) is Open image in new window .

Here, we take Open image in new window , thus Open image in new window is defined for Open image in new window .

## 3. Non-Dirichlet to Affine

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

Theorem 3.1.

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

Proof.

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

where Open image in new window , and recall Open image in new window .

Note that for Open image in new window , this corresponds to the results in [1] for Open image in new window .

Theorem 3.2.

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

Proof.

which is of the form (3.14), where Open image in new window , Open image in new window , and Open image in new window .

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

Remark 3.3.

In Theorems 3.1 and 3.2, we have taken Open image in new window to be a solution of (2.2) for Open image in new window with Open image in new window less than the least eigenvalue of (2.2), (3.2), and (3.13) such that Open image in new window in Open image in new window . We assume that Open image in new window 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 Open image in new window If Open image in new window , then we must have that either Open image in new window and Open image in new window or Open image in new window and Open image in new window . The first case of Open image in new window is not possible since Open image in new window and Open image in new window , Open image in new window , which implies that Open image in new window in particular for Open image in new window . For the second case, we get Open image in new window and Open image in new window which is not possible. Thus for Open image in new window , there are no allowable values of Open image in new window .

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

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 Open image in new window , and thus we may assume that Open image in new window .

Since Open image in new window , we consider the two cases, Open image in new window and Open image in new window .

Now if Open image in new window , then the numerator of Open image in new window is strictly negative. Thus, in order that Open image in new window , we require that the denominator is strictly negative, that is, Open image in new window . So either Open image in new window and Open image in new window or Open image in new window and Open image in new window . As Open image in new window , we obtain that either Open image in new window and Open image in new window or Open image in new window and Open image in new window . These are the same conditions as we had on Open image in new window for Open image in new window . Thus, the sign of Open image in new window does not play a role in finding the allowable values of Open image in new window which ensure that Open image in new window , 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 Open image in new window is a solution to (2.2) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.2) with one of the four sets of boundary conditions above, such that Open image in new window in the given interval Open image in new window , takes Open image in new window obeying (2.2) to Open image in new window obeying (2.5).

- (i)where Open image in new window and(4.6)
where Open image in new window with Open image in new window .

- (ii)
Open image in new window obeying (4.2) transforms to Open image in new window obeying (4.5) and (3.14).

- (iii)
Open image in new window obeying(4.3) transforms to Open image in new window obeying (3.3) and (4.6).

- (iv)
Open image in new window obeying (4.4) transforms to Open image in new window obeying (3.3) and (3.14).

The next theorem, shows that the boundary value problem given by Open image in new window obeying (2.2) together with any one of the four types of boundary conditions in the above theorem has Open image in new window eigenvalues as a result of the eigencondition being the solution of an Open image in new window th order polynomial in Open image in new window . 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 Open image in new window obeying (2.2) together with any one of the four types of boundary conditions given by (4.1) to (4.4) has Open image in new window eigenvalues.

Proof.

where Open image in new window and Open image in new window are real constants, that is, a first order polynomial in Open image in new window .

where again Open image in new window are real constants, that is, a quadratic polynomial in Open image in new window .

where Open image in new window , Open image in new window and Open image in new window , Open image in new window are real constants, that is, an Open image in new window th and an Open image in new window th order polynomial in Open image in new window , respectively.

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

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

where Open image in new window and Open image in new window are real constants, that is, a first order polynomial in Open image in new window .

where again Open image in new window , Open image in new window are real constants, that is, a quadratic polynomial in Open image in new window .

where Open image in new window , Open image in new window and Open image in new window , Open image in new window are real constants, thereby giving an Open image in new window th and an Open image in new window th order polynomial in Open image in new window , respectively.

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

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

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

Theorem 4.3.

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

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window , Open image in new window are real constants, that is, an Open image in new window th, Open image in new window th, and Open image in new window th order polynomial in Open image in new window , respectively.

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

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

where Open image in new window , Open image in new window , and Open image in new window are real constants.

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window are real constants.

where all the coefficients of Open image in new window are real constants.

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

which is an Open image in new window th order polynomial in Open image in new window and thus has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.5) with transformed boundary conditions (iv), that is, (3.3) and (3.14), has Open image in new window 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 Open image in new window are the eigenvalues of the original boundary value problem with corresponding eigenfunctions Open image in new window , then Open image in new window are eigenfunctions of the corresponding transformed boundary value problem with eigenvalues Open image in new window . Since we know that the transformed boundary value problem has Open image in new window eigenvalues, it follows that Open image in new window constitute all the eigenvalues of the transformed boundary value problem, see [1].

## 5. Affine to Non-Dirichlet

we can transform Open image in new window obeying affine Open image in new window -dependent boundary conditions to Open image in new window obeying non-Dirichlet boundary conditions.

Theorem 5.1.

where Open image in new window and Open image in new window .

Proof.

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

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

Remark 5.2.

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

hence Open image in new window .

Thus, Open image in new window , that is, Open image in new window .

that is, Open image in new window .

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 Open image in new window eigenvalues.

- (1)
boundary conditions (a) transform to Open image in new window and Open image in new window ;

- (2)
boundary conditions (b) transform to Open image in new window and (3.13);

- (3)
boundary conditions (c) transform to (3.2) and Open image in new window ;

- (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 Open image in new window each have Open image in new window eigenvalues. In particular, if Open image in new window are the eigenvalues of (2.5), (a) ((b), (c), (d), resp.) with eigenfunctions Open image in new window , then Open image in new window and Open image in new window are eigenfunctions of (2.2), (1) ((2), (3), (4), resp.) with eigenvalues Open image in new window . Since we know that these boundary value problems have Open image in new window eigenvalues, it follows that Open image in new window 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.

## Notes

### 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.

## References

- 1.Currie S, Love A:
**Transformations of difference equations I.***Advances in Difference Equations*2010,**2010:**-22.Google Scholar - 2.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

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