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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 145967, 11 pages
http://dx.doi.org/10.1155/2013/145967
Research Article

On the Estimations of the Small Periodic Eigenvalues

Department of Mathematics, Dogus University, Acıbadem, Kadiköy, 81010 Istanbul, Turkey

Received 15 April 2013; Revised 13 June 2013; Accepted 16 June 2013

Academic Editor: Ferhan M. Atici

Copyright © 2013 Seza Dinibütün and O. A. Veliev. 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.

Abstract

We estimate the small periodic and semiperiodic eigenvalues of Hill's operator with sufficiently differentiable potential by two different methods. Then using it we give the high precision approximations for the length of th gap in the spectrum of Hill-Sehrodinger operator and for the length of th instability interval of Hill's equation for small values of Finally we illustrate and compare the results obtained by two different ways for some examples.

1. Introduction

Let and be the operators generated in by the differential expression with the periodic and semiperiodic boundary conditions, respectively, where is a real periodic function with period . The eigenvalues of and for are and for , respectively. All eigenvalues of and , except , are doubled. The eigenvalues of the operators and , called periodic and semiperiodic eigenvalues, are denoted by and for , respectively, where [1, see page 27]. The spectrum of the operator generated in by (1) and the boundary conditions is the union of the periodic and semiperiodic eigenvalues, that is, since (5) holds if and only if either (2) or (3) holds [1, see page 33].

The spectrum of the operator generated in by (1) consists of the intervals for . Moreover, these intervals are the closure of the stable intervals of equation The intervals for are the gaps in the spectrum. These intervals with are the instable intervals of (7) [1, see pages 32 and 82]. The length of th gap in the spectrum of (the length of th instability interval of (7)) is Therefore the estimations of the periodic and semiperiodic eigenvalues are also the investigations of the spectrum of and of the stable intervals of (7).

In this paper we gave the estimations for the small periodic and semiperiodic eigenvalues when the real periodic potential belongs to the Sobolev space with . These assumptions on the potential imply that where Without loss of generality, it is assumed that .

It is wellknown that (see [2]) To give a subtle estimate for the eigenvalues , we write the potential in the form where The inequality in (9) implies that Hence, by the perturbation theory (see [2]) we have Therefore to estimate and we can investigate the eigenvalues of the operator and then use (8) and (15).

In the literature, there are a lot of studies about numerical estimation of the periodic and semiperiodic eigenvalues by using the finite difference method, finite element method, Prüfer transformations, and shooting method. Let us recall some of them. Andrew considered the computations of the eigenvalues by using finite element method [3] and finite difference method [4]. Then these results have been extended by Condon [5] and by Vanden Berghe et al. [6]. Ji and Wong used Prüfer transformation and shooting method in their studies [79]. Malathi et al. [10] used shooting technique and direct integration method for computing eigenvalues of periodic Sturm-Liouville problems.

We consider the small periodic and semiperiodic eigenvalues by other methods. First, in Section 2, we obtain an approximation of the eigenvalues for , where is the positive integer for determination of the error in estimations, by using the method of the paper [11], where the asymptotic formulas for the eigenvalues and eigenfunctions of the periodic boundary value problems were obtained. Then, in Section 3, using it and considering the matrix form of we give an approximation with very small errors for all small periodic and semiperiodic eigenvalues. Finally, we apply these investigations to get approximations order , and for the first eigenvalues of the operator with potentials , and , respectively, and give a comparison between the approximated eigenvalues obtained by the different ways.

2. On Applications of the Asymptotic Methods

In this and next sections, for simplicity of the notation, is denoted by . By (11)–(13) To get the subtle estimations for , that is, to observe the influence of the trigonometric polynomial to the eigenvalue of , we use the formula obtained from the equation by multiplying , where is the eigenfunction corresponding to the eigenvalue and denote inner product and norm in .

Introduce the notation Using this notation and (13) in (17) we get In (20) replacing by and then iterating it times, as in the paper [11], were done; we obtain where under assumption that for . Now using (21), estimating and , we prove the following,

Theorem 1. Let be a positive integer. If the conditions hold, then the eigenvalue of the operator satisfieswhere , , and are defined in (19) and (13).

Proof. Since we have . This with (16), (19), and (26) implies that for ; that is, assumption (25) holds. Therefore we can use (21).
Now we estimate and . First let us estimate . Since by Schwarz inequality we have This with (23) and (29) implies that Hence by definition of (see (19)) we have
Now we estimate . Arguing as in the proof of (29) we get Therefore using (17) we get This with Parseval’s equality implies that Hence at least one of the inequalities holds. If the first inequality holds, then dividing both sides of (21) by and using (23), (32) we obtain the proof of (27) and (28). If the second inequality holds, then instead of (21) using taking into account that and arguing as in the first case we get the proof in the second case. Theorem is proved.

Now using (27) let us show that is close to the root of the equation where

Theorem 2. Let be a positive integer satisfying Then for all and from the inequality where holds, and (39) has a unique solution on . Moreover and the length of th gap in the spectrum of (the length of th instability interval of (7)) satisfies

Proof. Let ,, be the first, second, and th summations in the right-hand side of (40). Then For , using (29) and (41), we get On the other hand This inequality with (47) and the inequality (see (19)) imply that In the same way we obtain for . Thus by the geometric series formula we have where is defined in (43), and by mean-value theorem (42) holds. Therefore by contraction mapping theorem (39) has a unique solution on .
Now let us prove (44). Let . Using the definition of and and then (40) we obtain and On the other hand by (51) we have for all . Therefore using the mean-value formula , and (52) we obtain This with (28) implies (44) for . In the same way we prove (44) for . Therefore (45) follows from (44). The theorem is proved.

Now let us approximate by fixed-point iteration Note that repeating the proof of (51) one can readily see that for all satisfying (41).

Theorem 3. For the sequence defined by (55) the estimations for hold, where satisfies (41), is defined in Theorem 2, and

Proof. It is clear and well known that if satisfies (42) then Therefore to prove (57) it is enough to show that where is defined in (58). By definition of and we have and by the mean-value theorem there exists such that These two equalities imply that This formula with (56) and (51) implies (60).

Thus by (44) and (57) we have the approximation for with the error

3. Estimation of the Small Eigenvalues

In this section we estimate the eigenvalues of the operator , for , by investigating the system of equations for , where and is the positive integer for determination of the error in estimation, the numbers and are defined in (19). The first, second, and th equations of (65) are obtained from (20) by taking , and , respectively, and by writing the terms with multiplicand for on the left-hand side and the terms with multiplicand for on the right-hand side.

To write (65) in the matrix form let us introduce the notations. Let be by matrix defined by for and if and all other entries of are zero. Since (see (9)), is a Hermitian (self-adjoint) matrix and its eigenvalues are real numbers. Denote the eigenvalues of by , where It is clear that since the diagonal elements of are for and the sum of the absolute values of the nondiagonal elements of each row is not greater than (see (19)). Let and be vectors of , where for and for . In this notation the system of (65) can be written in the matrix form

First we prove that for , that is, the right-hand side of (71), is small (see Lemma 4). Then using it we prove that the th eigenvalue of the operator is close to the th eigenvalue of the matrix (see Theorem 6).

Lemma 4. If and , then

Proof. First we prove (72) for positive . The proof for negative is similar. One can readily see from the estimations (27), (28) for , (56), and (66) that if , then Using (74) and taking into account the condition on and we obtain for . On the other hand iterating (20) times we get Therefore arguing as in the proof of (32) we get for ; that is, (72) is proved.
Now we prove (73). By definition of the left-hand side of (73) can be written in the form where In (72) replacing by one can readily see that Using this in (78) we obtain which implies (73), since the series in the right-hand side of (81) is a geometric series with first term and factor .

Note that (72) and (73) imply the following inequalities. By (70) and (72) and by the definition of we have Besides using (73) and Parseval’s equality (35) we obtain

Let be orthonormal system of eigenvectors of the matrix : where , and denotes the inner product in as well as in . Denote by the diagonal matrix with diagonal elements for . The eigenfunctions of corresponding to the eigenvalues are and , where , and for all . Multiplying both sides of (85) for by we get where if . Instead of (20) using (87) and repeating the proof of (72) we obtain that if and , then

To prove the main result of the paper we use the following.

Lemma 5. Let and . Then for and for .

Proof. Since is an orthonormal basis in we have Using this in (71) we get Multiplying both sides by we obtain On the other hand using the definition , (82), and (88) we get for all . This with (93) implies (89).
By Schwarz inequality and (83) we have for all and . Therefore (90) follows from (93).

Introduce the notation Here and are elements of , and Using equality (35) and the definition of and one can easily verify that and are the orthonormal systems in .

Now we are ready to prove the following main result.

Theorem 6. If then the inequality holds for all , where and are defined in (65) and (19).

Proof. Suppose to the contrary and without loss of generality that (98) does not hold for some . Then either or . Let us consider the case . Then and hence by (89) for all . It implies that for . On the other hand from Parseval’s equality (91) we have
Now we are going to get a contradiction by proving that the left-hand side of (101) is greater than the right-hand side of (101). Using (84), the definition of , and the conditions on one can easily verify that To estimate the right-hand side of (101) we write it as , where Using (100) and taking into account that and hence we obtain
Now let us estimate . Using (99), (69), and then the inequality we obtain for and . Therefore this, (90), and the definition imply that
Now let us estimate . Using (97) and the Bessel inequality for the elements for with respect to the orthonormal systems of we obtain The inequalities (104)–(107) show that the right side of (101) is less than , which contradicts (102). In the same way we investigate the case . The theorem is proved.

4. Examples and Conclusion

In this section we illustrate the results of Sections 2 and 3 for the following examples. Let the potential for of the operator have the form that is, for and for , where is defined in (9). Note that the operator is a famous Mathieu operator. By (19) and (108), and . For , the constant or has the values of , respectively. The fixed point approximations determined in (55), where is defined by (40) with , of the eigenvalues of the operators for are given in Tables 1, 2, and 3, respectively. Moreover, the estimations of the error (see (64)) and the length of the th gap (see (45)) are also given in Tables 1, 2, and 3.

tab1
Table 1: Estimations for .
tab2
Table 2: Estimations for .
tab3
Table 3: Estimations for .

The method of Section 3 gives high precision results for the calculation of the small eigenvalues. Let us illustrate it by using formula (98) for the first eigenvalues ,,,,, of the operators for . It means that the number in (98) is (see the first sentence of Section 3). To find an approximation with error of order for the eigenvalues of we take . Therefore for the potential , where , the number is and the number of equations in (65) is . The matrices of (65) corresponding to the potentials and denoted by are of order , and , respectively. The approximate eigenvalues of the matrices are given in Table 4. By (98) the eigenvalues are very close to the eigenvalues of the operator . One can readily see from (98) that the approximation of by the eigenvalues is arbitrary small if is a large number and is a small number. If the potential is smooth function, then the number is a small number (see (13) and (19)), and hence (98) gives better approximations for smooth potentials. Moreover if is a small number, that is, the number of summand of (see (108)) is small, then we can choose so that the order of the matrix is not a large number while the approximation (98) is a very small number. By formula (98) , where , for the potentials , and is not greater than respectively. Thus in Section 3 there are the following observations to be considered. Instead of the matrices of order investigating a little big matrices, namely, matrices of order , and , we find an approximation of order , and for the first eigenvalues of , and , respectively. Moreover this approach is applicable for the trigonometric polynomial potentials and for the sufficiently differentiable periodic potentials.

tab4
Table 4: Approximation of eigenvalues.

The estimations of the lengths of the gaps are given in Table 5. It is known that [12] for large the behavior of is sensitive to smoothness properties of the potential . If is times differentiable, then . If is analytic function, then for some positive . For the Mathieu operator the following asymptotic formula holds: . Thus for large the length of the th gap is a very small number. Table 5 confirms this result for large (see for ). Moreover Table 5 shows that these results continue to hold for . Since for the small values of ( the asymptotic formulas do not give any information, we cannot compare the theoretical results with the results in Table 5. Note that in Tables 4 and 5 the eigenvalues and the lengths of the gaps are computed using Matlab. In Table 4 this program transects to 14 figures, because this accuracy is acceptable for estimations of the eigenvalues. However, we compute the lengths of the gaps without transaction, since (as it is noted above) for large the theoretical results give the estimations of with very high accuracy.

tab5
Table 5: Approximation of the lengths of the gaps.

It is natural and well known that for large eigenvalues the asymptotic method gives us approximations with smaller errors. Since the method of Section 3 gives high precision results for the small eigenvalues and gaps (see Tables 4 and 5), the comparison of the Tables 15, where we estimate the eigenvalues and gaps by the methods of Sections 2 and 3, respectively, for the potential (108), shows that the results of the asymptotic method given in Tables 13 are not precise for the small eigenvalues.

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