On a non-definite Sturm-Liouville problem in the two turning point case - analysis and numerical results

In this paper, we study the non-definite Sturm-Liouville problem comprising of a regular Sturm-Liouville equation and Dirichlet boundary conditions on a closed interval. We consider the case in which the weight function changes sign twice in the given interval of definition. We give detailed numerical results on the spectrum of the problem, from which we verify various results on general non definite Sturm-Liouville problems. We also present some theoretical results which support the numerical results. Some numerical results seem to be in contrast with the results that are so far obtained in the case where the weight function changes sign once. This leads to more open questions for future studies in this particular area.


Introduction
The regular Sturm-Liouville problem involves finding the values of a parameter λ (generally complex) for which the equation  We pronounce that the strong interest of this field during all these years is that this theory is important in Applied Mathematics, where SL problems occur very commonly.
The differential equations considered here arise directly as mathematical models of motion according to Newton's law, but more often as a result of using the method of separation of variables to solve the classical partial differential equations of physics, such as Laplace's equation, the heat equation, and the wave equation, (see e.g [1]). Let (1) be written as Then, the problem consisting of (4) and the boundary conditions , wu u is definite, then the problem is called non-definite (indefinite).
In this paper our focus is on a non-definite Sturm-Liouville problem in which the weight function ( ) w x has two turning points in the interval of definition.

The Non-Definite (or Indefinite) Case
Here we give a summary on the non-definite case, detailed literature can be found in the papers [3]- [9], etc, and the references there in. In the non-definite case the spectrum is discrete, always consists of a doubly infinite sequence of real eigenvalues, and has at most a finite and even number of non-real eigenvalues (necessarily occurring in complex conjugate pairs). Remark 1. If the problem consisting of the equation and the boundary conditions (2)-(3) has N distinct negative eigenvalues, then the number of distinct pairs of non-real eigenvalues of the problem (1)-(3) cannot exceed N.
For more details on remark 1, we refer the interested reader to the papers [2] (Theorem 4.2.1), [3] (Theorem 2), [10] (Corollary 1.7), and the references there. In the non-definite case, as Richardson [4] puts it, the march of the zeros is not monotone with λ ∈  (in contrast with the left-and right-definite cases). In fact there may be a range of values of λ such that as λ increases, the number of zeros first decreases, then increases, then decreases and finally increases, the minimum number being a positive integer. As a result the eigenfunction corresponding to the eigenvalue 0 λ can have any number of zeros in ( ) , a b in contrast with the definite case, that is to say, a non-definite Sturm-Liouville problem will tend not to have a real ground state (positive eigenfunction). In relation to this behaviour of the real spectrum of the non-definite Sturm-Liouville problem, Mingarelli [6] defines two types of indexes which are due to Richardson [4] and Haupt [11].
We note that H n λ λ + + < . We can interpret λ + as the smallest number such that the real eigenvalues greater than λ + behave as in a "typical" Sturm-Liouville problem, that is, an eigenvalue is uniquely associated with its oscillation number, and λ − is interpreted similarly [7]. We note that in the right-definite case, λ λ + − = = As Jabon and Atkinson [7] rightly put it, in the non-definite case, the determination of these numbers is a very significant problem. In relation to corollary 1, we state the following theorem which is due to Richardson [4], see also the papers [3] [6]. 1) Estimate the oscillation numbers R n + and H n + in terms of the given data , , , p q w etc.
2)Estimate the eigenvalues R n λ + and H n λ + in terms of the given data.
3) Give sufficient conditions for the existence of at least one non-real eigenvalue. 4) Estimate the real and imaginary parts of non-real eigenvalues.

5) Is Richardson's oscillation theorem for non-real eigenfunctions true in general?
6) To what extent is Richardson's theorem for non-real eigenfunctions true?
The following is a brief list of part of the work done towards answering some of the questions raised above.
1) In the one-turning point case for w, Atkinson and Jabon, [7] obtain upper bound for λ + and lower bound for λ − .
2) In the two-turning point case for w, Kikonko and Mingarelli [8] obtain upper bound on λ + .

5) On
Richardson's Oscillation theorem, numerical results in the conference paper [18] indicated that the interlacing property fails in the two-turning point case and no non-real eigenfunction vanished inside the given interval of definition at least for the values of 0 q that were considered then.
The main motivation for this paper is the results obtained from the important paper [7] in which the Authors considered a special indefinite (non-definite) problem in which the weight function ( ) w x has one turning point in the interval ( ) 1) The interlacing property which holds in the one-turning point case does not hold in the two turning-point case in general.
2) The real and imaginary parts of any non-real eigenfunction corresponding to a non-real eigenvalue either have the same number of zeros in the interval ( )

Main Results
Here we consider the Dirichlet problem where we assume, without loss of generality, that 0, . In this case, the forms (5) and (6) and fixing the values of A, B and C to be  (8) Figure   1.
The summary of the results are shown in Table 1 and Table 2. Table 1 brings out the difference between the number of zeros of real and imaginary parts of the non-real eigenfunctions corresponding to non-real eigenvalues of the problem (8)- (9). The results in this table are complemented by the results shown in Figure 1 which shows that the number of zeros of the real and imaginary parts of the non-real eigenfunctions are either equal or differ by two. Figure 1 also shows that the interlacing property of

Discussion
From Figure 3, we see that the spectrum is made up of an infinite number of real eigenvalues and a finite number of non-real eigenvalues for each value of 0 q considered. That the number of non-real eigenvalues of problem (8)-(9) is finite, is not a surprise because this is expected, by remark 1. It can be seen from the graphs of the eigenfunctions that generally oscillation numbers decrease as the parameter value increases, but then oscillations will stabilize and the usual oscillation theorem eventually holds. This leads to the estimation of λ + , R n + , and H n + . We also observe However, for some values of λ a few oscillations are expected in the first and last intervals. This is so because in some cases, 0 q can be so large that 0 0. q w λ − > For example in Figure 2, eigenfunctions corresponding to the first three positive eigenvalues have at least one zero in the first and third intervals.
Generally speaking, the number of non-real eigenvalues seems to increase with . This was not one of the observation in the paper [18] in which we only considered generally smaller values of 0 q .

Conclusions
In this paper, we undertook a numerical study of the non-real eigenfunctions and eigenvalues of a non-definite Sturm-Liouville problem with two turning points, paralleling the study in [7] in the case of one turning point. Our ultimate goal was to examine the behavior of the eigenfunctions, both real and non-real, of this non-definite Sturm-Liouville problem.
One of the interesting observations was that the zeros of the real and imaginary parts of a non-real eigenfunction interlace in some subintervals of ( ) Furthermore, the number of zeros of the real part of each of the non-real eigenfunctions considered is greater (by two) than the number of zeros of the imaginary part in some cases, while in other cases, the number of zeros of the real part is equal to that of the imaginary part of a non-real eigenfunction corresponding to a non-real eigenvalue. Also this may be a consequence of a more general theorem which we don't know, so then, we have a third interesting open question for future research.
Summing up, we mean that the research initiated in [18] and presented in detail in this paper has implied a number of new interesting open questions of both theoretical and practical importance.