Transformations of Difference Equations II
© S. Currie and A. D. Love. 2010
Received: 13 April 2010
Accepted: 6 September 2010
Published: 14 September 2010
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.
This paper is structured as follows.
The relevant results from , 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.
where and are constants, see . Without loss of generality, by a shift of the spectrum, we may assume that the least eigenvalue, , of (1.1), (2.1) is .
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.
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.
Note that for , this corresponds to the results in  for .
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.
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 .
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 .
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.
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).
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.
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.
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 .
5. Affine to Non-Dirichlet
Note that the case of , that is, a non-Dirichlet boundary condition, gives , that is, which corresponds to the results obtained in .
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.
To summarise, we have the following.
non-Dirichlet and non-Dirichlet, that is, (4.5) and (4.6);
non-Dirichlet and affine, that is, (4.5) and (3.14);
affine and non-Dirichlet, that is, (3.3) and (4.6);
affine and affine, that is, (3.3) and (3.14).
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.
non-Dirichlet at the initial point and affine at the terminal point;
affine at the initial point and non-Dirichlet at the terminal point;
affine at the initial point and at the terminal point.
Dirichlet at the initial point and non-Dirichlet at the terminal point;
non-Dirichlet at the initial point and Dirichlet at the terminal point;
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.
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.
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.