Transformations of Difference Equations II

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

This paper continues the work done in [1], where we considered a weighted second-order difference equation of the following form:

(1.1)

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

Consider the second-order difference equation (1.1) for with boundary conditions

(2.1)

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 .

We recall the following important results from [1]. The mapping defined for by , where is the eigenfunction of (1.1), (2.1) corresponding to the eigenvalue , produces the following transformed equation:

(2.2)

where

(2.3)

Moreover, obeying the boundary conditions (2.1) transforms to obeying the Dirichlet boundary conditions as follows:

(2.4)

Applying the mapping given by for , where is a solution of (2.2) with , where is less than the least eigenvalue of (2.2), (2.4), such that for all , results in the following transformed equation:

(2.5)

where, for ,

(2.6)

Here, we take , thus is defined for .

In addition, obeying the Dirichlet boundary conditions (2.4) transforms to obeying the non-Dirichlet boundary conditions as follows:

(2.7)

where

(2.8)

3. Non-Dirichlet to Affine

In this section, we show how obeying the non-Dirichlet boundary conditions (3.2), (3.13) transforms under the following mapping:

(3.1)

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.

Under the transformation (3.1), obeying the boundary conditions

(3.2)

for , transforms to obeying the boundary conditions

(3.3)

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.

Evaluating (2.5) at for and using (3.3) gives the following:

(3.4)

Also from (3.1) for and , we obtain the following:

(3.5)

Substituting (3.2) into the above equation yields

(3.6)

Thus, (3.4) becomes

(3.7)

This may be slightly rewritten as follows

(3.8)

Also from (2.2), with , together with (3.2), we have the following:

(3.9)

Subtracting (3.9) from (3.8) and using the fact that results in

(3.10)

Equating coefficients of on both sides gives the following:

(3.11)

and equating coefficients of on both sides gives the following:

(3.12)

where , and recall .

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

Theorem 3.2.

Under the transformation (3.1), satisfying the boundary conditions

(3.13)

for , transforms to obeying the boundary conditions

(3.14)

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.

Evaluating (3.1) at and gives the following:

(3.15)

(3.16)

By considering satisfying (2.2) at , we obtain that

(3.17)

Substituting (3.17) into (3.16) gives the following:

(3.18)

Now using (3.13) together with (3.15) yields

(3.19)

which in turn, by substituting into (3.13), gives the following:

(3.20)

Thus, by putting (3.19) and (3.20) into (3.18), we obtain

(3.21)

The equation above may be rewritten as follows:

(3.22)

Now, since is a solution to (2.2) for , we have that

(3.23)

Substituting (3.23) into (3.22) gives the following:

(3.24)

Setting yields

(3.25)

Hence,

(3.26)

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.

If where is a solution to (2.2) for with less than the least eigenvalue of (2.2), (3.2), and (3.13) and in the given interval , then the values of which ensure that in (3.3), that is, which ensure that in (3.3) is a Nevanlinna function are

(3.27)

Proof.

From Theorem 3.1, we have that

(3.28)

Assume that , then to ensure that we require that either and or and . For the first case, since , we get and . For the second case, we obtain and , which is not possible. Thus, allowable values of for are

(3.29)

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.

If where is a solution to (2.2) for with less than the least eigenvalue of (2.2), (3.2), and (3.13), and in the given interval , then

(3.30)

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 .

Assume that , then the numerator of is strictly positive. Thus, to ensure that the denominator must be strictly positive, that is, . So either and or and . Since , we have that either and or and . Thus, if , that is, , we get

(3.31)

and if , that is, , we get

(3.32)

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.

Assume that satisfies (2.2). Consider the following four sets of boundary conditions:

(4.1)

(4.2)

(4.3)

(4.4)

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).

In addition,

• (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.

Since obeys (2.2), we have that, for ,

(4.7)

So setting , in (4.7), gives the following:

(4.8)

For the boundary conditions (4.1) and (4.2), we have that giving

(4.9)

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

Also in (4.7) gives that

(4.10)

Substituting in for , from above, we obtain

(4.11)

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

Thus, by an easy induction, we have that

(4.12)

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

Now, (4.1) gives , that is,

(4.13)

So our eigencondition is given by

(4.14)

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.

Next, (4.2) gives , so

(4.15)

from which we obtain the following eigencondition:

(4.16)

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.

Now for the boundary conditions (4.3) and (4.4), we have that , thus (4.8) becomes

(4.17)

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

Using and from above, we can show that can be written as the following:

(4.18)

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

Thus, by induction,

(4.19)

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

Now, (4.3) gives , that is,

(4.20)

So our eigencondition is given by

(4.21)

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.

Lastly, (4.4) gives , that is,

(4.22)

from which we obtain the following eigencondition:

(4.23)

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 .

Since obeys (2.5), we have that, for ,

(4.24)

For the transformed boundary conditions in (i) and (ii) of Theorem 4.1, we have that (4.5) is obeyed, and as in Theorem 4.2, we can inductively show that

(4.25)

and also by [1], we can extend the domain of to include if necessary by forcing (4.6) and then

(4.26)

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

Now for (i), the boundary condition (4.6) gives the following:

(4.27)

Therefore, the eigencondition is

(4.28)

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.

Also, for (ii), from the boundary condition (3.14), we get

(4.29)

Therefore, the eigencondition is

(4.30)

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.

Putting in (4.24), we get

(4.31)

For the boundary conditions in (iii) and (iv), we have that (3.3) is obeyed, thus,

(4.32)

where , , and are real constants.

Putting in (4.24), we get

(4.33)

which, by using (3.3) and , can be rewritten as follows:

(4.34)

where , , , , and are real constants.

Thus, inductively we obtain

(4.35)

Also, by [1], we can again extend the domain of to include , if needed, by forcing (4.6), thus,

(4.36)

where all the coefficients of are real constants.

The transformed boundary conditions (iii) mean that (4.6) is obeyed, thus, our eigencondition is

(4.37)

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.

Also, the transformed boundary conditions in (iv) give (3.14) which produces the following eigencondition:

(4.38)

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

In this section, we now show that the process in Section 3 may be reversed. In particular, by applying the following mapping:

(5.1)

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

Theorem 5.1.

Consider the boundary value problem given by satisfying (2.5) with the following boundary conditions:

(5.2)

(5.3)

The transformation (5.1), for , where is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue , yields the following equation:

(5.4)

where, for ,

(5.5)

In addition, obeying (5.2) and (5.3) transforms to obeying the non-Dirichlet boundary conditions

(5.6)

(5.7)

where and .

Proof.

The fact that , obeying (2.5), transforms to , obeying (5.4), was covered in [1, conclusion]. Now, is defined for and is extended to by (5.2). Thus, is defined for giving that (5.4) is valid for . For and , (5.1) gives the following:

(5.8)

(5.9)

Setting in (2.5) gives the following:

(5.10)

which by using (5.2) becomes

(5.11)

Since is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue , we have that

(5.12)

and hence

(5.13)

Substituting (5.11) and (5.13) into (5.8) and using (5.2), we obtain

(5.14)

Since , everything can be written over the common denominator . Taking out and simplifying, we get

(5.15)

Thus,

(5.16)

Substituting (5.2) into (5.9) gives the following:

(5.17)

Hence, by putting (5.16) into (5.17), we get

(5.18)

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.

Evaluating (5.4) at and using (5.7) give the following:

(5.19)

Using (5.1) with and together with (5.3), we obtain

(5.20)

Substituting the above two equations into (5.19) yields

(5.21)

Since is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue we have that . Thus, the above equation can be simplified to

(5.22)

Also (2.5) evaluated at for together with (5.3) gives

(5.23)

Adding (5.22) to (5.23) and using the fact that yields

(5.24)

By substituting in for and , it is easy to see that all the terms cancel out. Next, we examine the coefficients of , and using , we obtain that the coefficient of is

(5.25)

which equals by (2.5) evaluated at . Thus, only the terms in remain. First, we note that by substituting in for , and we get

(5.26)

Thus, equating coefficients of gives the following:

(5.27)

Since , we can divide and solve for to obtain

(5.28)

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

If we set , with a solution of (2.2) for where less than the least eigenvalue of (2.2), (3.2), and (3.13) and in the given interval , then is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue . To see that satisfies (2.5), see [1, Lemma ] with, as previously, , and now . Then, by construction, obeys (5.2). We now show that obeys (5.3). Let ,

(5.29)

Now is a solution of (2.2) for , thus,

(5.30)

Remark 5.2.

For , , , , and as above, the transformation (5.1), in Theorem 5.1, results in the original given boundary value problem. In particular, we obtain that in Theorem 5.1   and , see [1, Theorem .2]. In addition,

(5.31)

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.

We now verify that . Let

(5.32)

Since , we obtain , and thus

(5.33)

Also, . Dividing through by and using together with gives the following:

(5.34)

Now,

(5.35)

and since satisfies (2.2) at for , we get

(5.36)

Thus, using (5.35) and (5.36), the numerator of is simplified to

(5.37)

The denominator of can be simplified using to

(5.38)

hence .

Finally, substituting in for , we obtain

(5.39)

Thus, , that is, .

Next, we show that . Recall that and

(5.40)

Let

(5.41)

Note that

(5.42)

and since satisfies (2.2) at for , we get

(5.43)

We now substitute in for and into the equation for and use (5.42) and (5.43) to obtain that

(5.44)

that is, .

To summarise, we have the following.

Consider obeying (2.5) with one of the following 4 types of boundary conditions:

• (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.

Now, the transformation (5.1), with an eigenfunction of (2.5) with boundary conditions (a) ((b), (c), (d), resp.) corresponding to the eigenvalue , transforms obeying (2.5) to obeying (2.2) and transforms the boundary conditions as follows:

• (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

To conclude, we outline (the details are left to the reader to verify) how the entire process could also be carried out the other way around. That is, we start with a second order difference equation of the usual form, given in the previous sections, together with boundary conditions of one of the following forms:

• (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.

We can then transform the above boundary value problem (by extending the domain where necessary, as done previously) to an equation of the same type with, respectively, transformed boundary conditions as follows:

• (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.