^{1}

^{1}

The following inverse problem is solved—given the eigenvalues and the potential b(n) for a difference boundary value problem with quadratic dependence on the eigenparameter, λ, the weights c(n) can be uniquely reconstructed. The investi-gation is inductive on m where represents the number of unit intervals and the results obtained depend on the specific form of the given boundary conditions. This paper is a sequel to [1] which provided an algorithm for the solution of an analogous inverse problem, where the eigenvalues and weights were given and the potential was uniquely reconstructed. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in [1], an additional spectrum is required more often than was the case in [1].

Inverse problems in general are important in modern-day mathematics as they appear in many situations in physics, engineering, biology and medicine. This paper deals with inverse problems pertaining to a special type of second order difference equation. A comprehensive introduction to difference equations can be found, for example, in [

This paper is a sequel to [

where

Given the weights and the eigenvalues for the above boundary value problem with

We now investigate the following inverse problem. Given a spectrum for a boundary value problem of the form (1), (2) and (3), with

The paper has the following structure. The proof of the above inverse problem is done inductively beginning with the cases of

As we do not have experimental data for the two examples presented in Section 3, the eigenvalues used are obtained by first solving the “forward” problem. Consequently, the theoretical results for certain of the inverse problems are then verified using these eigenvalues.

An important result concerning the number of eigenvalues associated with a par- ticular boundary value problem was proved in [

Theorem 1.1. Consider the boundary value problem given by Equation (1) for

1)

2)

3)

(Note that the number of unit intervals considered is

In this section we investigate how to reconstruct the difference boundary value problem using a given spectrum or spectra. That is, how does the given spectrum/spectra, to- gether with the potential function

Consider (1) with boundary conditions

The cases of

In certain instances it is necessary to consider a second boundary value problem in order to obtain unique results. The second problem will be given by (1) with boundary conditions of the form

The case

The cases for

As the inequalities for

Again, we will split the inverse problem for

Theorem 3.1. Consider the boundary value problems (1), (4), (5) and (1), (6), (7) where

1)

2)

The boundary value problems corresponding to 1) and 2) have three eigenvalues, say

Proof. From Theorem 1.1, for both cases 1) and 2), it is clear that the boundary value problems each have three eigenvalues.

1) Assume

Next, evaluating (1) at

Also, at

Then applying (5) gives the equation

which on simplification yields a polynomial in

This can be rewritten in the form

The eigencondition is given by

For (1), (6) and (7) we obtain the same third order equation as (12) with ^{~}) versions.

In addition, the eigencondition in this case is given by

Equating coefficients of

and similarly considering the coefficients of

Solving the six simultaneous equations gives

2) Suppose that

This can be rewritten as a cubic polynomial i.e. in the form (12) where

Again for (1), (6) and (7) we obtain the same cubic polynomial as above with ^{~}) versions. The eigenconditions are given by (13) and (14) thus, Equations (15) and (16) hold. Hence, we can again solve the six simultaneous equations uniquely for

Theorem 3.2. Consider the boundary value problem (1), (4) and (5) where

1)

2)

3)

4)

The boundary value problem corresponding to any of the four cases above has four eigenvalues, say

Proof. This uses the procedure outlined in Theorem 3.1 above. It is similar to the proof of Theorem 3.2 in [

and similarly for

Theorem 3.3. Consider the boundary value problem (1), (4) and (5) where

1)

2)

3)

Given the five eigenvalues

Proof. In all three cases, starting with

The associated eigencondition is also a fifth order polynomial given by

where

By equating relevant coefficients of powers of

Theorem 3.4. Assume that we have the boundary value problem (1), (4) and (5) with

Proof. As per usual we start the evaluation of (1) at

Example 1

To illustrate part (3) of Theorem 3.3, suppose that

Clearly, it is seen that,

Example 2

Assume that we are given eigenvalues

To illustrate Theorem 3.2(3), suppose also that

Note that the two boundary value problems above are of the same form i.e. they have exactly the same equation and their boundary conditions are of the same type. In the first problem,

As mentioned previously, the case of

Theorem 4.1. For

1) if

2) if

Proof. Follows as in ( [

Remark: 1) It is not possible for the number of eigenvalues of (1), (4) and (5) to be less than m as this would imply a Dirichlet boundary condition at

2) It should be noted that because there are more weights than potentials, two spectra are required more often than in [

We thank the Editor and the referee for their comments. Research of S. Currie is supported by NRF grant no. IFR2011040100017. This support is greatly appreciated.

Currie, S. and Love, A. (2016) Inverse Problems for Difference Equations with Quadratic Eigenparameter Dependent Boundary Conditions-II. Advances in Pure Mathematics, 6, 625-632. http://dx.doi.org/10.4236/apm.2016.610051