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

Seza Dinibütün and O. A. Veliev

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 [7–9]. 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 satisfies

where

, , 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.

Table 1: Estimations for .

49.0119073043627
0.00401827341683563
0.00803652968036530

64.0090356900908
0.00232226049016466
0.00464451589853519

81.0070967373201
0.00146120590904089
0.00292241001412498

100.005724155838
0.00097836132370372
0.00195672191528545

400.001412301984
8.57660779334148 × 10⁻⁵
0.00017153215300668

900.000626190365
2.27805363772165 × 10⁻⁵
4.55610726195539 × 10⁻⁵

1600.00035193858
9.11289409047171 × 10⁻⁶
1.82257881647412 × 10⁻⁵

2500.00022515394
4.5213654576927 × 10⁻⁶
9.04273091219341 × 10⁻⁶

3600.00015632421
2.56272510680566 × 10⁻⁶
5.12545021275656 × 10⁻⁶

4900.00011483597
1.59021161389524 × 10⁻⁶
3.18042322750835 × 10⁻⁶

6400.0000879141
1.05364682405463 × 10⁻⁶
2.10729364800086 × 10⁻⁶

8100.0000694591
7.33717636691826 × 10⁻⁷
1.46743527333693 × 10⁻⁶

10000.0000562596
5.31248113844258 × 10⁻⁷
1.06249622766647 × 10⁻⁶

Table 2: Estimations for .

169.006553875546
0.00801822430426367
0.01603644646924830

196.005629484083
0.00602192413268590
0.01204384713096120

225.004888933687
0.00463706113393842
0.00927412163367219

256.004286247051
0.00364633797625785
0.00729267558174552

289.003789043447
0.00291892692052321
0.00583785361582058

324.003373962035
0.00237284215158748
0.00474568416174256

361.003023794203
0.00195492184884340
0.00390984360625575

400.002725629827
0.00162966444959707
0.00325932883855288

900.001203843083
0.00040657655861401
0.00081315311464415

1600.00067569654
0.00015796542624476
0.00031593085219360

2500.00043201403
7.70627570578593 × 10⁻⁵
0.00015412551405901

3600.00029984729
4.32014439412344 × 10⁻⁵
8.64028878675565 × 10⁻⁵

4900.00022022411
2.66004761436181 × 10⁻⁵
5.32009522823765 × 10⁻⁵

6400.00016857341
1.75241067230997 × 10⁻⁵
3.50482134443501 × 10⁻⁵

8100.00013317449
1.21491644560633 × 10⁻⁵
2.42983289113352 × 10⁻⁵

10000.0001078601
8.76569198293195 × 10⁻⁶
1.75313839654927 × 10⁻⁵

Table 3: Estimations for .

361.004488989457
0.012018209724884
0.024036418816389

400.004042369632
0.00990095572697077
0.0198019110415735

900.001776635742
0.00230548774143664
0.00461097546712649

1600.00099552468
0.00086829674995455
0.00173659349819394

2500.00063601104
0.00041615591363515
0.00083231182695096

3600.00044125198
0.00023064366100756
0.00046128732193269

4900.00032399850
0.00014088338405301
0.00028176676807951

6400.00024796876
9.22660434495073 × 10⁻⁵
0.00018453208688902

8100.00019587583
6.36768120642807 × 10⁻⁵
0.00012735362412432

10000.0001586304
4.57782012510395 × 10⁻⁵
9.15564025001002 × 10⁻⁵

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.

Table 4: Approximation of eigenvalues.

−0.455138604105
−0.451676027152
−0.4539320948685

−0.110248816992
−0.040158274572
−0.0204737818081

1.859108072514
1.4456177812459
1.3907354889190

3.917024772994
2.8976658743702
2.9541319115098

4.371300982731
5.1886431499537
4.8580498527548

9.047739259808
8.9161585304864
7.9082824512658

9.078368847202
9.4153327308285
10.2941738497520

16.032970081406
16.0004107071615
15.9213717462580

16.033832340360
16.1585649096071
16.3957158213096

25.020840823290
25.0389311983095
24.9848629686203

25.020854345449
25.0538295076160
25.1789211080558

36.014289910633
36.0293767228453
36.0144251509371

36.014290046045
36.0319035321757
36.0877507661928

49.010418249424
49.0218195042565
49.0311600838136

49.010418250365
49.0219701639618
49.0394601884444

64.007937189247
64.0164674336750
64.0248999242659

64.007937189258
64.0164851040169
64.0271961781896

81.006250326633
81.0128685694864
81.0198291868669

81.006250326634
81.0128693419217
81.0203601164022

100.005050675157
100.010339593273
100.015987594137

100.005050675158
100.010339662550
100.016034084442

400.001253135321
400.002520531313
400.003809046181

400.001253135326
400.002520531318
400.003809046182

900.000556173742
900.001115142518
900.001678193187

900.000556173751
900.001115142519
900.001678193192

1600.00031269547
1600.00062627292
1600.00094113218

1600.00031269548
1600.00062627292
1600.00094113219

2500.00020008004
2500.00040052089
2500.00060148494

2500.00020008004
2500.00040052089
2500.00060148495

3600.00013892748
3600.00027802883
3600.00041738205

3600.00013892749
3600.00027802885
3600.00041738205

4900.00010206165
4900.00020421710
4900.00030650836

4900.00010206165
4900.00020421711
4900.00030650836

6400.00007813720
6400.00015632939
6400.00023460110

6400.00007813721
6400.00015632940
6400.00023460110

8100.00006173602
8100.00012350634
8100.00018532630

8100.00006173602
8100.00012350635
8100.00018532634

10000.00005000500
10000.00010003250
10000.00015009260

10000.00005000500
10000.00010003260
10000.00015009260

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.

Table 5: Approximation of the lengths of the gaps.

1.96935688950626
1.48577605581811
1.41120927072708

0.45427620973738
2.29097727558352
1.90391794124493

0.03062958739405
0.49917420034206
2.38589139848613

0.00086225895372
0.15815420244566
0.47434407505152

1.35221586674561 × 10⁻⁵
0.01489830930653
0.19405813943552

1.35412271617952 × 10⁻⁷
0.00252680933036
0.07332561525572

9.41085431804822 × 10⁻¹⁰
0.00015065970523
0.00830010463081

1.09992015495664 × 10⁻¹¹
1.76703419043633 × 10⁻⁵
0.00229625392370

5.82645043323282 × 10⁻¹³
7.72435271301219 × 10⁻⁷
0.00053092953526

1.22213350550737 × 10⁻¹²
6.92769646093439 × 10⁻⁸
4.64903052659338 × 10⁻⁵

5.11590769747272 × 10⁻¹²
4.88853402202949 × 10⁻¹²
9.09494701772928 × 10⁻¹³

8.64019966684282 × 10⁻¹²
4.54747350886464 × 10⁻¹³
4.43378667114303 × 10⁻¹²

3.41060513164848 × 10⁻¹²
3.63797880709171 × 10⁻¹²
1.02318153949454 × 10⁻¹¹

3.18323145620525 × 10⁻¹²
6.82121026329696 × 10⁻¹²
5.45696821063757 × 10⁻¹²

7.27595761418343 × 10⁻¹²
1.90993887372315 × 10⁻¹¹
4.09272615797818 × 10⁻¹²

3.63797880709171 × 10⁻¹²
5.45696821063757 × 10⁻¹²
2.72848410531878 × 10⁻¹²

6.3664629124105 × 10⁻¹²
5.45696821063757 × 10⁻¹²
1.81898940354586 × 10⁻¹²

1.81898940354586 × 10⁻¹²
6.3664629124105 × 10⁻¹²
4.09272615797818 × 10⁻¹¹

1.09139364212751 × 10⁻¹¹
2.91038304567337 × 10⁻¹¹
0

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 1–5, 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 1–3 are not precise for the small eigenvalues.

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