- Research
- Open Access
Difference boundary value problem hierarchies and the forward Crum transformation
- Sonja Currie1Email author and
- Anne D Love1
https://doi.org/10.1186/1687-1847-2013-311
© Currie and Love; licensee Springer. 2013
- Received: 7 June 2013
- Accepted: 11 October 2013
- Published: 8 November 2013
Abstract
This paper uses the forward transformation defined in Currie and Love (Adv. Differ. Equ. 2010:947058, 2010) to develop a hierarchy for difference boundary value problems with eigenparameter-dependent boundary conditions. In particular, we show how various sets of these boundary conditions transform under this forward transformation. The resulting hierarchy is then illustrated in a tabular form where the number of eigenvalues for each of the problems is also given. In addition, we point out certain analogies to the work done in Currie and Love (Bound. Value Probl. 2011:743135, 2011) where the reverse Crum-type transformation was used to establish a hierarchy of difference boundary value problems.
MSC:39A05, 34K10, 65Q10, 65Q30.
Keywords
- difference equations
- hierarchies
- boundary value problems
- transformations
- Nevanlinna
- eigenvalues
1 Introduction
One of the main reasons for studying the Darboux-Crum transformation on a discrete second-order difference equation is its relevance to quantum mechanics in the computation of discrete energy levels and corresponding eigenfunctions. This follows from the fact that a second-order difference equation can be factorised as a product of two Crum-type transformations, see [1].
This paper develops an alternative hierarchy to that given in [2] for difference boundary problems by using the forward Crum-type transformation that was given initially in [1]. Our interest in this alternate hierarchy is its value in the study of the associated inverse problem which is our current project. Although the results appear similar to those in [2–4], it should be noted that they do not follow directly from these papers and as such, it is necessary that these new results be proved as shown in Sections 3.1 and 3.2.
are imposed at the initial and terminal endpoints.
We prove how (1.2) transforms under transformation (2.1). It is necessary to consider the cases of and separately in order to establish our results (see Theorem 3.5). The majority of the work carried out in this paper pertains to the transformation of (1.3) under (2.1) (see Theorems 3.2, 3.3 and 3.4). Here we consider the cases and separately because when it is necessary to then further consider independently the cases of and . It should be noted that when the sign of b is immaterial. Interestingly, in Theorem 3.4, the case when both and is unique in that the interval of the transformed boundary value problem shrinks by one unit. Furthermore, a comparison of the original and transformed boundary value problems and their corresponding eigenvalues is given in Tables 1 and 2 in Section 4. In addition, the analogies between the hierarchy presented in this paper and that found in [2–4] are mentioned.
In mathematics, the difference equation is often used as a means to study its continuous counterpart, the differential equation. However, difference equations are interesting and useful in their own right. Difference equations or recurrence relations describe a situation where there is a discrete sequence of entities, each of which either gives an input to its successor, or perhaps interacts with both its neighbors.
with and , were once used as a model for the growth of rabbit populations.
For an application of (1.1) with variable coefficients together with eigenparameter-dependent boundary conditions, we turn our attention to quantum physics. Here being a variable indicates that the medium or space considered is non-uniform. The coefficient , which may or may not vary, is the discrete analogue of the potential in the continuous case. The significance of the eigenparameter-dependent boundary conditions is that the boundary of the space responds to the energy of the system, i.e. is not fixed, where the eigenparameter λ represents the energy level (up to a scaling factor). In particular, in the one-dimensional discrete model of an electron in an atom, λ gives the energy state of the electron.
It should be noted that Harmsen and Li, in [6], study discrete Sturm-Liouville problems where the eigenparameter appears linearly in the boundary conditions. These results are then extended by the authors in [7] to difference boundary value problems where the spectral parameter appears quadratically in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.
Difference equations are also being studied using Lie group theory. This usually involves a symmetry-based approach to solving a given ordinary difference equation by considering the structure of the solution set. A symmetry is a continuous group which leaves the system of equations invariant. For example, if a one-dimensional difference equation admits a symmetry, then it becomes a linear equation and an analytic solution may be obtained as described in [8] and [9]. Hydon in [10] develops a method for finding one-parameter Lie groups of symmetries to achieve successive reductions of order. The difference equation can be completely solved provided there are a sufficient number of symmetries. Levi and Winternitz in [11] show how the apparent mismatch between discrete equations and continuous symmetries can be resolved. Their approach is either to use generalised symmetries on the solutions of the difference equations which leave the lattice invariant or, alternatively, restrict to point symmetries which allow them to transform the lattice.
Section 2 provides the preliminary results from [1–4]. Section 3 pertains to the transformation of boundary conditions (1.2) and (1.3) under mapping (2.1) as discussed above, while Section 4 provides the reader with a tabularized comparison of various sets of boundary conditions and eigenvalues for the original boundary value problem with those of the transformed boundary value problem.
2 Preliminaries
This section provides a recap of the necessary results from [1–4].
Consider equation (1.1) for . Note that the values of and are given by the boundary conditions.
where, throughout this paper, is a solution to (1.1) for such that for all .
- (i)
If does satisfy both given boundary conditions (1.2) and (1.3), then the spectral parameter must be shifted so that the original boundary value problem has least eigenvalue precisely 0.
- (ii)
If does not obey one or both given boundary conditions (1.2) and (1.3), then the spectral parameter of the original given boundary value problem must first be shifted so as to ensure that all the eigenvalues are nonnegative.
- (iii)
Once the required shifts above have been made, is taken to be a solution to the shifted equation, which we may assume is given again by (1.1) for .
From [1] we have the following theorem.
- (A)If , are positive Nevanlinna functions, then(2.5)
- (B)If(2.6)
- (C)If(2.7)
Positive Nevanlinna function.
The notation used will follow that in [2], namely will denote a Nevanlinna function where:
s is the number of terms in the sum;
The number of eigenvalues that a particular boundary value problem has depends on the form of the boundary conditions and the theorem below was proved in [4].
- (i)
eigenvalues if ,
- (ii)
eigenvalues if and ,
- (iii)
eigenvalues if .
(Note that the number of unit intervals considered is .)
3 Transformation of boundary conditions
In this section we investigate how obeying general eigenparameter-dependent boundary conditions of the form given in (1.2) and (1.3) transforms under (2.1) to obeying corresponding boundary conditions which depend on the spectral parameter. Choosing various zero or non-zero values for a, b, g, h gives rise to the different results which are proved below. By considering the number of zero’s and poles (singularities) of the various Nevanlinna functions involved, we obtain the precise form of these transformed boundary conditions.
3.1 Boundary condition at the terminal end point
The following lemma could be considered as an analogy to Lemma 3.1 in [2].
as required. □
It is still necessary to show that is a negative Nevanlinna function of the correct form. In the following theorem, we consider the case where and in (3.1).
- (1)If does not obey (3.20) for , then obeys(3.21)
- (2)If does obey (3.20), then obeys(3.22)
where .
Since is a positive Nevanlinna function, it has a graph of the form shown in Figure 1 where , , , such that , and for . Note that since , does not exist.
If does not obey (3.20), then the zeros of are the poles of , that is, ’s together with . Since there is the same number of ’s as there are ’s, it follows that in (3.24).
This results in two cases: either and , which is not possible as , or we must have and . This in turn means that for it follows by Nevanlinna property (C) that is a negative Nevanlinna function of the form (3.21) as required.
If does obey (3.20) for , then , i.e. . Thus, one of ’s, , is equal to 0.
where γ, Γ and Ω are as previously defined. Once again, for W to be a positive Nevanlinna function, we require that . Hence, by Nevanlinna result (C), may be written as a negative Nevanlinna function of the form (3.22). □
- (A)
If does not obey (3.28) for , then obeys a boundary condition of the form
- (i)(3.29)
- (ii)(3.30)
- (B)
If does obey (3.28) for , then obeys a boundary condition of the form
- (i)(3.31)
- (ii)(3.32)
where , i.e. , , , are negative Nevanlinna functions.
where . Here ’s correspond to which are the poles of (i.e. ’s) and .
The graph of is given by Figure 1 with , , , and for . Note again that since , does not exist.
Clearly, the gradient of at is positive and the number of ’s is the same as the number of ’s, thus in (3.35), .
where , and . Since , it follows that .
as required by (3.29).
that is, (3.30) holds.
Therefore, the discontinuity at is removable giving that the number of non-removable ’s is one less than the number of ’s. This implies that in (3.35) the number of terms in the sum equals s so, relabeling if necessary, we can set .
The remainder of the results are obtained in exactly the same manner as in the case where does not obey boundary condition (3.28) with s being replaced by , see (3.36) and the subsequent calculations. Thus we obtain the following:
which is a negative Nevanlinna function of the correct form, i.e. equation (3.31).
where . That is, we obtain (3.32). □
- (A)If does not obey (3.37) for , then obeys a boundary condition of the form(3.38)
- (B)If does obey (3.37) for , then obeys a boundary condition of the form(3.39)
where , i.e. , are negative Nevanlinna functions.
where ’s correspond to , i.e. the singularities of (3.46).
As is a positive Nevanlinna function, it has a graph of the form shown in Figure 1 where , , , , for and for .
Observe that the gradient of at is positive for all .
are the poles of (i.e. ’s) and . It is evident that the number of ’s is one more than the number of ’s, thus in (3.47), .
where , and making the right-hand side a negative Nevanlinna function, i.e. we obtain (3.38).
If does obey (3.37) for , then . Thus, one of ’s, , is equal to 0. As in Theorems 3.2 and 3.3, it can be shown that the singularity at is removable giving that the number of non-removable ’s is equal to the number of ’s.
where ϕ is as defined above, and , i.e. a negative Nevanlinna function of the form (3.39). □
3.2 Boundary condition at the initial end point
For transformation (2.1), the theorem below could be considered analogous to Theorems 3.2 and 3.3 given in [2] for the reverse transformation.
- (A)
If does not obey (3.50) for , then obeys a boundary condition of the form
- (i)(3.51)
- (ii)(3.52)
- (B)
If does obey (3.50) for , then obeys a boundary condition of the form
- (i)(3.53)
- (ii)(3.54)
where , , , , i.e. , , , are positive Nevanlinna functions.
where corresponds to the singularities of (3.58), that is, where .
If in (3.50), then the graph has the form shown in Figure 1 where , , , such that , and for . Note that since , does not exist.
If does not obey (3.50) when , then the zeros of are the poles of , that is, ’s together with . Since there is the same number of ’s as there are ’s, it follows that in (3.60).
which is of the form required by (3.51).
so that once again the discontinuity at zero has been removed. This means that the number of non-removable discontinuities, i.e. ’s, is one less than the number of ’s, that is, in (3.60) put .
If, in equation (3.50), , then the graph of has the form shown in Figure 1 where , , , , for and for .
It is evident that the gradient of is positive at ’s.
where .
□
4 Conclusion
Boundary condition ( 3.50 ) with
Original BVP: (1.1) with bc’s … | Trans. BVP: (2.2) with bc’s … | |
---|---|---|
1 | (3.20) and (3.50) with g = 0 | (3.21) and (3.51) |
z does not obey (3.20) or (3.50) | s + 1 + p + 1 + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. one extra eigenvalue 0 | |
2 | (3.20) and (3.50) with g = 0 | (3.22) and (3.51) |
z obeys (3.20) but not (3.50) | s + p + 1 + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. same eigenvalues | |
3 | (3.20) and (3.50) with g = 0 | (3.21) and (3.53) |
z obeys (3.50) but not (3.20) | s + 1 + p + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. same eigenvalues | |
4 | (3.20) and (3.50) with g = 0 | (3.22) and (3.53) |
z obeys both (3.20) and (3.50) | s + p + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. one less eigenvalue 0 | |
5 | (3.28) and (3.50) with g = 0 | (3.29) and (3.51) |
z does not obey (3.28) or (3.50), | s + p + 1 + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one extra eigenvalue 0 | |
6 | (3.28) and (3.50) with g = 0 | (3.31) and (3.51) |
z obeys (3.28) but not (3.50), | s − 1 + p + 1 + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
7 | (3.28) and (3.50) with g = 0 | (3.29) and (3.53) |
z obeys (3.50) but not (3.28), | s + p + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
8 | (3.28) and (3.50) with g = 0 | (3.31) and (3.53) |
z obeys both (3.28) and (3.50), | s − 1 + p + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one less eigenvalue 0 | |
9 | (3.28) and (3.50) with g = 0 | (3.30) and (3.51) |
z does not obey (3.28) or (3.50), | s + 1 + p + 1 + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one extra eigenvalue 0 | |
10 | (3.28) and (3.50) with g = 0 | (3.32) and (3.51) |
z obeys (3.28) but not (3.50), | s + p + 1 + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
11 | (3.28) and (3.50) with g = 0 | (3.30) and (3.53) |
z obeys (3.50) but not (3.28), | s + 1 + p + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
12 | (3.28) and (3.50) with g = 0 | (3.32) and (3.53) |
z obeys both (3.28) and (3.50), | s + p + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one less eigenvalue 0 | |
13 | (3.37) and (3.50) with g = 0 | (3.38) and (3.51) |
z does not obey (3.37) or (3.50) | s + p + 1 + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. one extra eigenvalue 0 | |
14 | (3.37) and (3.50) with g = 0 | (3.39) and (3.51) |
z obeys (3.37) but not (3.50) | s − 1 + p + 1 + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. same eigenvalues | |
15 | (3.37) and (3.50) with g = 0 | (3.38) and (3.53) |
z obeys (3.50) but not (3.37) | s + p + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. same eigenvalues | |
16 | (3.37) and (3.50) with g = 0 | (3.39) and (3.53) |
z obeys both (3.37) and (3.50) | s − 1 + p + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. one less eigenvalue 0 |
Boundary condition ( 3.50 ) with
Original BVP: (1.1) with bc’s … | Trans. BVP: (2.2) with bc’s … | |
---|---|---|
1 | (3.20) and (3.50) with g>0 | (3.21) and (3.52) |
z does not obey (3.20) or (3.50) | s + 1 + p + 1 + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. one extra eigenvalue 0 | |
2 | (3.20) and (3.50) with g>0 | (3.22) and (3.52) |
z obeys (3.20) but not (3.50) | s + p + 1 + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. same eigenvalues | |
3 | (3.20) and (3.50) with g>0 | (3.21) and (3.54) |
z obeys (3.50) but not (3.20) | s + 1 + p + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. same eigenvalues | |
4 | (3.20) and (3.50) with g>0 | (3.22) and (3.54) |
z obeys both (3.20) and (3.50) | s + p + m − 1 eigenvalues | |
s + p + m eigenvalues | i.e. one less eigenvalue 0 | |
5 | (3.28) and (3.50) with g>0 | (3.29) and (3.52) |
z does not obey (3.28) or (3.50), | s + p + 1 + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one extra eigenvalue 0 | |
6 | (3.28) and (3.50) with g>0 | (3.31) and (3.52) |
z obeys (3.28) but not (3.50), | s − 1 + p + 1 + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
7 | (3.28) and (3.50) with g>0 | (3.29) and (3.54) |
z obeys (3.50) but not (3.28), | s + p + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
8 | (3.28) and (3.50) with g>0 | (3.31) and (3.54) |
z obeys both (3.28) and (3.50), | s − 1 + p + m + 1 eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one less eigenvalue 0 | |
9 | (3.28) and (3.50) with g>0 | (3.30) and (3.52) |
z does not obey (3.28) or (3.50), | s + 1 + p + 1 + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one extra eigenvalue 0 | |
10 | (3.28) and (3.50) with g>0 | (3.32) and (3.52) |
z obeys (3.28) but not (3.50), | s + p + 1 + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
11 | (3.28) and (3.50) with g>0 | (3.30) and (3.54) |
z obeys (3.50) but not (3.28), | s + 1 + p + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. same eigenvalues | |
12 | (3.28) and (3.50) with g>0 | (3.32) and (3.54) |
z obeys both (3.28) and (3.50), | s + p + m eigenvalues | |
s + p + m + 1 eigenvalues | i.e. one less eigenvalue 0 | |
13 | (3.37) and (3.50) with g>0 | (3.38) and (3.52) |
z does not obey (3.37) or (3.50) | s + p + 1 + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. one extra eigenvalue 0 | |
14 | (3.37) and (3.50) with g>0 | (3.39) and (3.52) |
z obeys (3.37) but not (3.50) | s − 1 + p + 1 + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. same eigenvalues | |
15 | (3.37) and (3.50) with g>0 | (3.38) and (3.54) |
z obeys (3.50) but not (3.37) | s + p + m − 1 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. same eigenvalues | |
16 | (3.37) and (3.50) with g>0 | (3.39) and (3.54) |
z obeys both (3.37) and (3.50) | s − 1 + p + m − 2 eigenvalues | |
s + p + m − 1 eigenvalues | i.e. one less eigenvalue 0 |
- (a)
If obeys the boundary conditions at both ends, then the transformed boundary value problem loses the eigenvalue 0 but retains the remaining eigenvalues from the original boundary value problem;
- (b)
If obeys the boundary condition at one end only, then the transformed boundary value problem will have exactly the same eigenvalues as the original boundary value problem;
- (c)
If does not obey any of the boundary conditions, then the transformed boundary value problem gains the eigenvalue 0, with corresponding eigenfunction , in addition to the eigenvalues of the original boundary value problem.
The above remark is consistent with the results obtained in [2, 3] and may be proved in the same way as [[2], Corollary 4.4], see also [3].
We conclude with the following observation. Since the reverse transformation used in [2] was at n and , whereas in this paper the forward transformation is at n and , the roles of the endpoints are in a sense ‘reversed’ which is well illustrated by the results obtained above.
Declarations
Acknowledgements
The first author is supported by NRF grant no. IFR2011040100017.
Authors’ Affiliations
References
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