# On the Estimations of the Small Periodic Eigenvalues

Authors: Seza Dinibütün; O. A. Veliev
Journal: Abstract and Applied Analysis
 (2013)

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## 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 ofnth gap in the spectrum of Hill-Sehrodinger operator and for the length of nth instability interval of  Hill's equation for small values of n. Finally we illustrate and compare the results obtained by two different ways for some examples.


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## Body

## 1. Introduction

LetP(q) and S(q) be the operators generated in L2[0,π] by the differential expression
(1)-y′′(x)+q(x)y(x)
with the periodic
(2)y(π)=y(0),y′(π)=y′(0)
and semiperiodic
(3)y(π)=-y(0),y′(π)=-y′(0)
boundary conditions, respectively, where q is a real periodic function with period π. The eigenvalues of P(q) and S(q) for q=0 are (2n)2 and (2n+1)2 for n∈ℤ, respectively. All eigenvalues of P(0) and S(0), except 0, are doubled. The eigenvalues of the operators P(q) and S(q), called periodic and semiperiodic eigenvalues, are denoted by λ2n and λ2n+1 for n∈ℤ, respectively, where
(4)λ0(q)<λ-1(q)≤λ1(q)<λ-2(q)≤λ2(q)<λ-3(q)≤λ3(q)<λ-4(q)≤λ4(q)⋯
[1, see page 27]. The spectrum σ(T(q)) of the operator T(q) generated in L2[0,2π] by (1) and the boundary conditions
(5)y(2π)=y(0),y′(2π)=y′(0)
is the union of the periodic and semiperiodic eigenvalues, that is,
(6)σ(P)={λ2n:n∈ℤ},σ(S)={λ2n+1:n∈ℤ},σ(T)={λn:n∈ℤ},
since (5) holds if and only if either (2) or (3) holds [1, see page 33].The spectrum of the operatorL(q) generated in L2(-∞,∞) by (1) consists of the intervals [λn-1(q),λ-n(q)] for n=1,2,…. Moreover, these intervals are the closure of the stable intervals of equation
(7)-y′′(x)+q(x)y(x)=λy(x).
The intervals (λ-n,λn) for n=1,2,… are the gaps in the spectrum. These intervals with (-∞,λ0) are the instable intervals of (7) [1, see pages 32 and 82]. The length of nth gap in the spectrum of L(q) (the length of (n+1)th instability interval of (7)) is
(8)γn(q)=:λn(q)-λ-n(q).
Therefore the estimations of the periodic and semiperiodic eigenvalues are also the investigations of the spectrum of L(q) 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 potentialq belongs to the Sobolev space W1k[0,π] with k>1. These assumptions on the potential q imply that
(9)q(x)=∑n∈ℤ‍qnei2nx,q-n=qn¯,|qn|≤r(2n)m,
where
(10)qn=(q,ei2nx)=∫0π‍q(x)e-i2nxdx,r=∫[0,π]‍|q(k)(x)|dx.
Without loss of generality, it is assumed that q0=0.It is wellknown that (see [2])
(11)|λn(q)-λn(0)|≤sup|q(x)|,λn(0)=n2,∀n∈ℤ.
To give a subtle estimate for the eigenvalues λn(q), we write the potential q in the form
(12)q(x)=p(x)+∑|n|>s‍qnei2nx,
where
(13)p(x)=∑n:|n|≤s‍qnei2nx.
The inequality in (9) implies that
(14)supx∈[0,π]|q(x)-p(x)|≤∑|n|>s‍|qn|≤r(k-1)(2s)k-1.
Hence, by the perturbation theory (see [2]) we have
(15)|λn(q)-λn(p)|≤r(k-1)(2s)k-1,|γn(q)-γn(p)|≤2r(k-1)(2s)k-1.
Therefore to estimate λn(q) and γn(q) we can investigate the eigenvalues λn(p) of the operator T(p) 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 Section2, we obtain an approximation of the eigenvalues λ±n(p) for n>ms, where m 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 t-periodic boundary value problems were obtained. Then, in Section 3, using it and considering the matrix form of T(p) we give an approximation with very small errors for all small periodic and semiperiodic eigenvalues. Finally, we apply these investigations to get approximations order 10-18,10-15, and 10-12 for the first 201 eigenvalues of the operator T with potentials p1(x)=2cos2x,p2(x)=2cos2x+2cos4x, and p3(x)=2cos2x+2cos4x+2cos6x, 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,λn(p) is denoted by λn. By (11)–(13)
(16)|λn-n2|≤sup|p(x)|≤∑n=-ss‍|qn|.
To get the subtle estimations for λn, that is, to observe the influence of the trigonometric polynomial p(x) to the eigenvalue n2 of T(0), we use the formula
(17)(λN-n2)(ΨN,einx)=(pΨN,einx)
obtained from the equation
(18)-ΨN′′(x)+p(x)ΨN(x)=λNΨN(x)
by multiplying einx, where ΨN is the eigenfunction corresponding to the eigenvalue λn;∥Ψn∥=1/π,(·,·) and ∥·∥ denote inner product and norm in L2[0,π].Introduce the notation(19)M=supx|p(x)|,c=∑n=-ss‍|qn|,Q=supn|qn|,XN,n=(ΨN,einx).
Using this notation and (13) in (17) we get
(20)(λN-n2)XN,n=∑k=-ss‍qkXN,n-2k.
In (20) replacing N by n and then iterating it m times, as in the paper [11], were done; we obtain
(21)(λn-n2)Xn,n=Am(λn,n)Xn,n+Rm+1(λn,n),
where
(22)Am(λn,n)=∑k=1m‍ak(λn,n),(23)ak(λn,n)=∑n1,n2,…,nk=-ss‍qn1qn2⋯qnkq-n1-n2-⋯-nk∏i=1,2,…,k[λn-(n-2n1-2n2⋯-2ni)2],Rm+1(λn,n)=∑n1,n2,…,nm+1=-ss‍×qn1qn2⋯qnmqnm+1Xn,n-2n1-2n2-⋯-2nm+1∏i=1,2,…m[λn-(n-2n1-2n2⋯-2ni)2],(24)nj≠0,∀j,∑j=1k‍nj≠0,∀k=1,2,…,m
under assumption that
(25)λn-(n-2n1⋯-2ni)2≠0
for i=1,2,…,m. Now using (21), estimating Xn,n and Rm+1, we prove the following,Theorem 1.
Letm be a positive integer. If the conditions
(26)|n|>ms,4(|n|-1)≥3M
hold, then the eigenvalue λn of the operator T(p) satisfies(27)λn=n2+∑k=1m‍∑n1,n2,…,nk=-ss‍×qn1qn2⋯qnkq-n1-n2-⋯-nk∏i=1,2,…k[λn-(n-2n1-2n2⋯-2ni)2]+αn,m,where
(28)|αn,m|≤2cm+1(4(|n|-1)-M)m;c, M, and p(x) are defined in (19) and (13).Proof.
Sinceq0=0 we have 0<|ni|≤s. This with (16), (19), and (26) implies that
(29)|λn-(n-2n1-2n2-⋯-2ni)2|≥|n2-(|n|-2)2|-M=4(|n|-1)-M≥2M>0
for i=1,2,…,m; that is, assumption (25) holds. Therefore we can use (21).
Now we estimateXn,n and Rm+1. First let us estimate Rm+1. Since ∥Ψn∥=1/π by Schwarz inequality we have
(30)|(Ψn(x),ei(n-2n1-2n2-⋯-2nm+1)x)|≤1.
This with (23) and (29) implies that
(31)|Rm+1|≤1(4(|n|-1)-M)m×∑n1,n2,…,nm+1=-ss‍|qn1qn2⋯qnmqnm+1|.
Hence by definition of c (see (19)) we have
(32)|Rm+1|≤cm+1(4(|n|-1)-M)m.
Now we estimateXn,n. Arguing as in the proof of (29) we get
(33)|λn-(n-2k)2|≥2M,∀k≠0,n.
Therefore using (17) we get
(34)∑k∈ℤ,k≠0,n‍|Xn,n-2k|2=∑k∈ℤ,k≠0,n‍|(Ψn,pei((n-2k))x)|2|λn-(n-2k)2|2≤M2(2M)2=14.
This with Parseval’s equality
(35)∑k∈ℤ‍|Xn,n-2k|2=∑k∈ℤ,‍|(Ψn,ei((n-2k))x)|2=1
implies that
(36)|Xn,n|2+|Xn,-n|2≥34.
Hence at least one of the inequalities
(37)|Xn,n|≥12,|Xn,-n|≥12
holds. If the first inequality holds, then dividing both sides of (21) by Xn,n and using (23), (32) we obtain the proof of (27) and (28). If the second inequality holds, then instead of (21) using
(38)(λn-(-n)2)Xn,-n=Am(λn,-n)Xn,-n+Rm+1(λn,-n),
taking into account that Am(λn,-n)=Am(λn,n) 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 λ±n is close to the root of the equation
(39)x=n2+f(x),
where
(40)f(x)=∑n1=-ss‍qn1q-n1x-(n-2n1)2+∑n1,n2=-ss‍qn1qn2q-n1-n2(x-(n-2n1)2)(x-(n-2n1-2n2)2)+…+∑n1,n2,…nm=-ss‍×((qn1qn2⋯qnmq-n1-n2-⋯-nm)×([x-(n-2n1)2][x-(n-2n1-2n2)2]⋯[x-(n-2n1-2n2-⋯-2nm)2][x-(n-2n1-2n2)2])-1((qn1qn2⋯qnmq-n1-n2-⋯-nm)).Theorem 2.
Letn be a positive integer satisfying
(41)n>ms,4(n-1)>M+2c.
Then for all x and y from [n2-M,n2+M] the inequality
(42)|f(x)-f(y)|<Kn|x-y|,
where
(43)Kn=Qc(4(n-1)-M)(4(n-1)-M-c)<12,
holds, and (39) has a unique solution rn on [n2-M,n2+M]. Moreover
(44)|λ±n-rn|<2cm+1(1-Kn)(4(n-1)-M)m
and the length γn of nth gap in the spectrum of L(p) (the length γn of (n+1)th instability interval of (7)) satisfies
(45)γn=λn-λ-n<4cm+1(1-Kn)(4(n-1)-M)m.Proof.
Letf1(x),f2(x),…,fm(x) be the first, second, and mth summations in the right-hand side of (40). Then
(46)f1′(x)=-∑k=-ss‍|qk2|(x-(n-2k)2)2.
For x∈[n2-M,n2+M], using (29) and (41), we get
(47)|x-(n-2k)2|≥4(n-1)-M>2c.
On the other hand
(48)∑k=-ss‍|qk2|≤Qc.
This inequality with (47) and the inequality Q≤c (see (19)) imply that
(49)|f1′(x)|≤Qc(4(n-1)-M)2<14.
In the same way we obtain
(50)|fk′(x)|≤Qck(4(n-1)-M)k+1<12k+1
for k=2,3,…. Thus by the geometric series formula we have
(51)|f′(x)|≤Kn<12,∀x∈[n2-M,n2+M],
where Kn is defined in (43), and by mean-value theorem (42) holds. Therefore by contraction mapping theorem (39) has a unique solution rn on [n2-M,n2+M].
Now let us prove (44). Let F(x)=x-n2-f(x). Using the definition of rn and F(x) and then (40) we obtain F(rn)=0 and
(52)|F(λn)-F(rn)|≤|αn,m|.
On the other hand by (51) we have |F′(x)|≥1-Kn for all x∈[n2-M,n2+M]. Therefore using the mean-value formula
(53)|F(λn)-F(rn)|=|F′(ζ)||λn-rn|,ζ∈[n2-M,n2+M], and (52) we obtain
(54)|λn-rn|≤|αn,m|1-Kn.
This with (28) implies (44) for λn. In the same way we prove (44) for λ-n. Therefore (45) follows from (44). The theorem is proved.Now let us approximatern by fixed-point iteration
(55)xn,0=n2,xn,1=n2+f(xn,0),…,xn,i=n2+f(xn,i-1).
Note that repeating the proof of (51) one can readily see that
(56)|f(λn)|≤Qc4(n-1)-M-c,|f(n2)|≤Qc4(n-1)-c
for all n satisfying (41).Theorem 3.
For the sequence{xn,i} defined by (55) the estimations
(57)|xn,i-rn|≤KniB
for i=1,2,3,… hold, where n satisfies (41), Kn is defined in Theorem 2, and
(58)B=|f(n2)|1-Kn≤Qc(1-Kn)(4(n-1)-c).Proof.
It is clear and well known that iff satisfies (42) then
(59)|xn,i-rn|≤Kni|xn,0-rn|.
Therefore to prove (57) it is enough to show that
(60)|xn,0-rn|≤B,
where B is defined in (58). By definition of rn and xn,0 we have
(61)rn-xn,0=f(rn)=f(rn)-f(xn,0)+f(n2),
and by the mean-value theorem there exists x∈[n2-M,n2+M] such that
(62)f(rn)-f(xn,0)=f′(x)(rn-xn,0).
These two equalities imply that
(63)(rn-xn,0)(1-f′(x))=f(n2).
This formula with (56) and (51) implies (60).Thus by (44) and (57) we have the approximation xn,i for λ±n with the error
(64)En,i=:|λ±n-xn,i|<2cm+1(1-Kn)(4(n-1)-M)m+KniB.


## 3. Estimation of the Small Eigenvalues

In this section we estimate the eigenvaluesλN of the operator T(p), for |N|≤l, by investigating the system of 2S+1 equations
(65)(λN-n2)XN,n-∑k:|k|≤s,|n-2k|≤S‍qkXN,n-2k=∑k:|k|≤s,|n-2k|>S‍qkXN,n-2k
for n=-S,-S+1,-S+2,…,S, where S=l+2rs and r is the positive integer for determination of the error in estimation,
(66)4(l-1)-M-c>max{c,2c2};
the numbers M and c are defined in (19). The first, second, and jth equations of (65) are obtained from (20) by taking n=-S,n=-S+1, and n=-S-1+j, respectively, and by writing the terms with multiplicand XN,n-2k for |n-2k|≤S on the left-hand side and the terms with multiplicand XN,n-2k for |n-2k|>S on the right-hand side.To write (65) in the matrix form let us introduce the notations. Let A be (2S+1) by (2S+1) matrix (ai,j) defined by
(67)ai,i=(-S-1+i)2,ai,i∓2k=q±k
for i=1,2,…,2S+1 and k=1,2,…s if |i∓2k|≤S and all other entries of A are zero. Since q-n=qn¯ (see (9)), A is a Hermitian (self-adjoint) matrix and its eigenvalues are real numbers. Denote the eigenvalues of A by μ0,μ-1,μ1,μ-2,μ2,…,μ-S,μS, where
(68)μ0≤μ-1≤μ1≤μ-2≤μ2≤⋯≤μ-S≤μS.
It is clear that
(69)|μ±n-n2|≤c,
since the diagonal elements of A are n2 for n=-S,-S+1,-S+2,…,S and the sum of the absolute values of the nondiagonal elements of each row is not greater than c (see (19)). Let XN=(XN,-S,XN,-S+1,…,XN,S) and R(λN)=(R-S,R-S+1,…,RS) be vectors of ℂ2S+1, where Rn=0 for |n|≤S-2s and
(70)Rn(λN)=∑k:|k|≤s,|n-2k|>S‍qkXN,n-2k
for S-2s<|n|≤S. In this notation the system of (65) can be written in the matrix form
(71)(λNI-A)XNT=RT(λN).First we prove thatXN,n for n=±(S+1),±(S+2),…,±(S+2s), that is, the right-hand side RT(λN) of (71), is small (see Lemma 4). Then using it we prove that the nth eigenvalue λn of the operator T(p) is close to the nth eigenvalue μn of the matrix A (see Theorem 6).Lemma 4.
If|N|≤l and l+2rs<|n|≤l+2(r+1)s, then
(72)|XN,n|≤cr+1(2l)r+1=:ε,(73)∑n:|n|>S‍|XN,n|2≤4sε2(2l)2((2l)2-c2)=4sc2r+2(2l)2r((2l)2-c2)=:δ.Proof.
First we prove (72) for positive n. The proof for negative n is similar. One can readily see from the estimations (27), (28) for m=2, (56), and (66) that if k≥l, then
(74)|λk-k2|≤|f(λk)|+|αk,2|≤Qc4(k-1)-M-c+2c3(4(k-1)-M)2<1.
Using (74) and taking into account the condition on N and n we obtain
(75)|λN-(n-2n1-⋯-2ni)2|≥|λN-(l+1)2|≥|λl-(l+1)2|>|l2-(l+1)2|-1≥2l
for |ni|≤s,i=0,1,…,r. On the other hand iterating (20) r times we get
(76)XN,n=∑n1,n2,…,nr=-ss‍×(⋯[λN-(n-2n1-⋯-2nr)2])-1(qn1qn2⋯qnr+1(ΨN,ei(n-2n1-⋯-2n2r+1)x))×([λN-n2][λN-(n-2n1)2]⋯[λN-(n-2n1-⋯-2nr)2])-1).
Therefore arguing as in the proof of (32) we get
(77)|XN,n|≤cr+1(2l)r+1
for l+2rs<|n|≤l+2(r+1)s; that is, (72) is proved.
Now we prove (73). By definition of S the left-hand side of (73) can be written in the form
(78)∑n:|n|>S‍|XN,n|2=∑k=r∞‍HN,k,
where
(79)HN,k=∑l+2ks<|n|≤l+2(k+1)s,‍|XN,n|2.
In (72) replacing r by k one can readily see that
(80)HN,k≤4sc2k+2(2l)2k+2.
Using this in (78) we obtain
(81)∑n:|n|>S‍|XN,n|2≤∑k=r∞‍4sc2k+2(2l)2k+2
which implies (73), since the series in the right-hand side of (81) is a geometric series with first term 4sε2 and factor c2/(2l)2.Note that (72) and (73) imply the following inequalities. By (70) and (72)
(82)|Rn(λN)|<cε,∀n:S-2s<|n|≤S,∀|N|≤l,
and by the definition of R(λN) we have
(83)∥R(λN)∥≤2cεs,∀|N|≤l.
Besides using (73) and Parseval’s equality (35) we obtain
(84)1-δ≤∑n=-SS‍|XN,n|2≤1,1-δ≤∥XN∥≤1,∀|N|≤l.Let{VnT:n=0,±1,±2,…,±S} be orthonormal system of eigenvectors of the matrix A:
(85)AVnT=μnVnT,
where 〈Vn,Vk〉=δn,k,Vn=(Vn,-S,Vn,-S+1,…,Vn,S)∈ℂ2S+1, and 〈·,·〉 denotes the inner product in ℂ2S+1 as well as in l2. Denote by D the (2S+1)×(2S+1) diagonal matrix with diagonal elements
(86)di=ai,i=(-S-1+i)2
for i=1,2,…,2S+1. The eigenfunctions of D corresponding to the eigenvalues n2 are e-n and en, where en=(en,-S,en,-S+1,…,en,S)T,en,n=1, and en,k=0 for all k≠n. Multiplying both sides of (85) for n=N by en we get
(87)(μN-n2)VN,n=∑k=-ss‍qkVN,n-2k,
where VN,n-2k=0 if |n-2k|>S. Instead of (20) using (87) and repeating the proof of (72) we obtain that if |N|≤l and |n|>S-2s, then
(88)|VN,n|≤cr(2l)r.To prove the main result of the paper we use the following.Lemma 5.
Letcn,j=〈XnT,VjT〉 and n=0,±1,±2,…,±l. Then
(89)|cn,j(μj-λn)|≤8slε2
for j=0,±1,±2,…,±l and
(90)|cn,j(μj-λn)|≤2cεs
for j=±(l+1),±(l+2),…,±S.Proof.
Since{Vj:j=0,±1,±2,…,±S} is an orthonormal basis in ℂ2S+1 we have
(91)XnT=∑j=-SS‍cn,jVjT,|Xk|2=∑j=-SS‍|ck,j|2.
Using this in (71) we get
(92)RT(λn)=(λnI-A)XnT=∑j=-SS‍(λncn,jVjT-A(cn,jVjT))=∑j=-SS‍cn,j(λn-μj)VjT.
Multiplying both sides by VjT we obtain
(93)cn,j(λn-μj)=〈RT(λn),VjT〉.
On the other hand using the definition RT(λn), (82), and (88) we get
(94)|〈RT(λn),VjT〉|≤4scεcr(2l)r=8slε2
for all n,j=0,±1,±2,…,±l. This with (93) implies (89).
By Schwarz inequality and (83) we have
(95)|〈RT(λn),VjT〉|≤2cεs
for all n=0,±1,±2,…,±l and j=0,±1,±2,…,±S. Therefore (90) follows from (93).Introduce the notation(96)Yn=(⋯Xn,-S-1,Xn,-S,Xn,-S+1,…,Xn,S,Xn,S+1,…),Uj=(⋯0,0,Vj,-S,Vj,-S+1,…,Vj,S,0,0,…).
Here Yn and Uj are elements of l2, and
(97)〈Yn,Uj〉=∑i=-∞∞‍Xn,iVj,i¯=∑i=-SS‍Xn,iVj,i¯=〈XnT,VjT〉=cn,j.
Using equality (35) and the definition of Yn and Uj one can easily verify that {Yn:n=0,±1,±2,…,±S} and{Un:n=0,±1,±2,…,±S} are the orthonormal systems in l2.Now we are ready to prove the following main result.Theorem 6.
Ifl>max{c2,2c,3s} then the inequality
(98)|λn-μn|≤8Ssc2r+2(2l)2r+1
holds for all n=0,±1,±2,…,±l, where S,r,l and c,s are defined in (65) and (19).Proof.
Suppose to the contrary and without loss of generality that (98) does not hold for some 0≤n≤l. Then either λn<μn-(8Ssc2r+2/(2l)2r+1) or λn>μn+(8Ssc2r+2/(2l)2r+1). Let us consider the case λn<μn-(8Ssc2r+2/(2l)2r+1). Then
(99)λk<μj-8Ssc2r+2(2l)2r+1,
and hence by (89) |ck,j|<1/2S for all k=0,±1,±2,…,±n,j=n,±(n+1),±(n+2)…,±l. It implies that
(100)|ck,n|2+∑j:n<|j|≤l‍|ck,j|2≤2l+1-2n4S2
for k=0,±1,±2,…,±n. On the other hand from Parseval’s equality (91) we have
(101)∑k=-nn‍|Xk|2=∑k=-nn‍∑j=-SS‍|ck,j|2.
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 δ, and the conditions on l one can easily verify that
(102)∑k=-nn‍|Xk|2≥2n+1-(2n+1)δ>2n+34.
To estimate the right-hand side of (101) we write it as S1+S2+S3, where
(103)S1=∑k=-nn‍(|ck,n|2+∑j:n<|j|≤l‍|ck,j|2),S2=∑k=-nn‍∑j=-nn-1‍|ck,j|2,S3=∑k=-nn‍(∑j:|j|>l‍|ck,j|2).
Using (100) and taking into account that (2l+1-2n)+(2n+1)≤2S and hence (2l+1-2n)(2n+1)≤S2 we obtain
(104)S1≤(2l+1-2n)(2n+1)4S2<14.
Now let us estimateS3. Using (99), (69), and then the inequality l>2c we obtain
(105)|λk-μj|>|μl-μj|>|j|
for k=0,±1,±2,…,±n and |j|>l. Therefore this, (90), and the definition ε imply that
(106)S3=∑k=-nn‍(∑j:|j|>l‍|ck,j|2)≤(2n+1)∑j:|j|>l‍(2cεsj)2<(2n+1)4sc2ε2l<14.
Now let us estimateS2. Using (97) and the Bessel inequality for the elements Ui for i=-n,-n+1,…,n-1 with respect to the orthonormal systems {Yn:n=0,±1,±2,…,±n} of l2 we obtain
(107)∑k=-nn‍|ck,i|2≤|Ui|2=1,S2=∑i=-nn-1‍∑k=-nn‍|ck,i|2≤2n.
The inequalities (104)–(107) show that the right side of (101) is less than 2n+(1/2), which contradicts (102). In the same way we investigate the case λn>μn+(8Ssc2r+2/(2l)2r+1). The theorem is proved.


## 4. Examples and Conclusion

In this section we illustrate the results of Sections2 and 3 for the following examples. Let the potential ps(x) for s=1,2,3 of the operator T(ps) have the form
(108)ps(x)=∑n=1s‍(ei2nx+e-i2nx)=∑n=1s‍2cos2nx;
that is,  qn=q-n=1 for 1≤n≤s and qn=q-n=0 for n>s, where qn is defined in (9). Note that the operator T(p1) is a famous Mathieu operator. By (19) and (108), Q=1 and M=c. For s=1,2,3 the constant M or c has the values of 2,4,6, respectively. The fixed point approximations xn,3 determined in (55), where f(x) is defined by (40) with m=3, of the eigenvalues λ±n of the operators T(ps) for s=1,2,3 are given in Tables 1, 2, and 3, respectively. Moreover, the estimations of the error En,3=|λ±n-xn,3| (see (64)) and the length γn of the nth gap (see (45)) are also given in Tables 1, 2, and 3.Table 1
Estimations forT(p1).
x
n
,
3
E
n
,
3
γ
n
n
=
7
49.0119073043627
0.00401827341683563
0.00803652968036530
n
=
8
64.0090356900908
0.00232226049016466
0.00464451589853519
n
=
9
81.0070967373201
0.00146120590904089
0.00292241001412498
n
=
10
100.005724155838
0.00097836132370372
0.00195672191528545
n
=
20
400.001412301984
8.57660779334148 × 10−5
0.00017153215300668
n
=
30
900.000626190365
2.27805363772165 × 10−5
4.55610726195539 × 10−5
n
=
40
1600.00035193858
9.11289409047171 × 10−6
1.82257881647412 × 10−5
n
=
50
2500.00022515394
4.5213654576927 × 10−6
9.04273091219341 × 10−6
n
=
60
3600.00015632421
2.56272510680566 × 10−6
5.12545021275656 × 10−6
n
=
70
4900.00011483597
1.59021161389524 × 10−6
3.18042322750835 × 10−6
n
=
80
6400.0000879141
1.05364682405463 × 10−6
2.10729364800086 × 10−6
n
=
90
8100.0000694591
7.33717636691826 × 10−7
1.46743527333693 × 10−6
n
=
100
10000.0000562596
5.31248113844258 × 10−7
1.06249622766647 × 10−6Table 2
Estimations forT(p2).
x
n
,
3
E
n
,
3
γ
n
n
=
13
169.006553875546
0.00801822430426367
0.01603644646924830
n
=
14
196.005629484083
0.00602192413268590
0.01204384713096120
n
=
15
225.004888933687
0.00463706113393842
0.00927412163367219
n
=
16
256.004286247051
0.00364633797625785
0.00729267558174552
n
=
17
289.003789043447
0.00291892692052321
0.00583785361582058
n
=
18
324.003373962035
0.00237284215158748
0.00474568416174256
n
=
19
361.003023794203
0.00195492184884340
0.00390984360625575
n
=
20
400.002725629827
0.00162966444959707
0.00325932883855288
n
=
30
900.001203843083
0.00040657655861401
0.00081315311464415
n
=
40
1600.00067569654
0.00015796542624476
0.00031593085219360
n
=
50
2500.00043201403
7.70627570578593 × 10−5
0.00015412551405901
n
=
60
3600.00029984729
4.32014439412344 × 10−5
8.64028878675565 × 10−5
n
=
70
4900.00022022411
2.66004761436181 × 10−5
5.32009522823765 × 10−5
n
=
80
6400.00016857341
1.75241067230997 × 10−5
3.50482134443501 × 10−5
n
=
90
8100.00013317449
1.21491644560633 × 10−5
2.42983289113352 × 10−5
n
=
100
10000.0001078601
8.76569198293195 × 10−6
1.75313839654927 × 10−5Table 3
Estimations forT(p3).
x
n
,
3
E
n
,
3
γ
n
n
=
19
361.004488989457
0.012018209724884
0.024036418816389
n
=
20
400.004042369632
0.00990095572697077
0.0198019110415735
n
=
30
900.001776635742
0.00230548774143664
0.00461097546712649
n
=
40
1600.00099552468
0.00086829674995455
0.00173659349819394
n
=
50
2500.00063601104
0.00041615591363515
0.00083231182695096
n
=
60
3600.00044125198
0.00023064366100756
0.00046128732193269
n
=
70
4900.00032399850
0.00014088338405301
0.00028176676807951
n
=
80
6400.00024796876
9.22660434495073 × 10−5
0.00018453208688902
n
=
90
8100.00019587583
6.36768120642807 × 10−5
0.00012735362412432
n
=
100
10000.0001586304
4.57782012510395 × 10−5
9.15564025001002 × 10−5The method of Section3 gives high precision results for the calculation of the small eigenvalues. Let us illustrate it by using formula (98) for the first 201 eigenvalues λ0,λ-1,λ1,λ-2,λ2,…,λ-100,λ100 of the operators T(ps) for s=1,2,3. It means that the number l in (98) is 100 (see the first sentence of Section 3). To find an approximation with error of order 10-18 for the eigenvalues of T(p1) we take r=5. Therefore for the potential ps(x), where s=1,2,3, the number S is l+2rs=100+10s and the number of equations in (65) is 2S+1=200+20s+1. The matrices of (65) corresponding to the potentials p1(x),p2(x),p3(x) and denoted by A1,A2,A3 are of order 221,241, and 261, respectively. The approximate eigenvalues μ0,μ-1,μ1,μ-2,μ2,…,μ-100,μ100 of the matrices A1,A2,A3 are given in Table 4. By (98) the eigenvalues μn are very close to the eigenvalues λn of the operator T(ps). One can readily see from (98) that the approximation |λn-μn| of λn by the eigenvalues μn is arbitrary small if r is a large number and c is a small number. If the potential q is smooth function, then the number c is a small number (see (13) and (19)), and hence (98) gives better approximations for smooth potentials. Moreover if s is a small number, that is, the number of summand of ps (see (108)) is small, then we can choose r so that the order of the matrix As is not a large number while the approximation (98) is a very small number. By formula (98) |λn-μn|, where n=0,±1,±2,…,±100, for the potentials p1(x),p2(x), and p3(x) is not greater than
(109)8×110×212(200)11=1162510-17,8×120×2×412(200)11=31907348632812500,8×130×3×612(200)11=2072619962500000000000000000,
respectively. Thus in Section 3 there are the following observations to be considered. Instead of the matrices of order 201 investigating a little big matrices, namely, matrices of order 221,241, and 261, we find an approximation of order 10-18,10-15, and 10-12 for the first 201 eigenvalues of T(p1),T(p2), and T(p3), respectively. Moreover this approach is applicable for the trigonometric polynomial potentials and for the sufficiently differentiable periodic potentials.Table 4
Approximation of eigenvalues.
p
1
p
2
p
3
λ
0
−0.455138604105
−0.451676027152
−0.4539320948685
λ
-
1
−0.110248816992
−0.040158274572
−0.0204737818081
λ
1
1.859108072514
1.4456177812459
1.3907354889190
λ
-
2
3.917024772994
2.8976658743702
2.9541319115098
λ
2
4.371300982731
5.1886431499537
4.8580498527548
λ
-
3
9.047739259808
8.9161585304864
7.9082824512658
λ
3
9.078368847202
9.4153327308285
10.2941738497520
λ
-
4
16.032970081406
16.0004107071615
15.9213717462580
λ
4
16.033832340360
16.1585649096071
16.3957158213096
λ
-
5
25.020840823290
25.0389311983095
24.9848629686203
λ
5
25.020854345449
25.0538295076160
25.1789211080558
λ
-
6
36.014289910633
36.0293767228453
36.0144251509371
λ
6
36.014290046045
36.0319035321757
36.0877507661928
λ
-
7
49.010418249424
49.0218195042565
49.0311600838136
λ
7
49.010418250365
49.0219701639618
49.0394601884444
λ
-
8
64.007937189247
64.0164674336750
64.0248999242659
λ
8
64.007937189258
64.0164851040169
64.0271961781896
λ
-
9
81.006250326633
81.0128685694864
81.0198291868669
λ
9
81.006250326634
81.0128693419217
81.0203601164022
λ
-
10
100.005050675157
100.010339593273
100.015987594137
λ
10
100.005050675158
100.010339662550
100.016034084442
λ
-
20
400.001253135321
400.002520531313
400.003809046181
λ
20
400.001253135326
400.002520531318
400.003809046182
λ
-
30
900.000556173742
900.001115142518
900.001678193187
λ
30
900.000556173751
900.001115142519
900.001678193192
λ
-
40
1600.00031269547
1600.00062627292
1600.00094113218
λ
40
1600.00031269548
1600.00062627292
1600.00094113219
λ
-
50
2500.00020008004
2500.00040052089
2500.00060148494
λ
50
2500.00020008004
2500.00040052089
2500.00060148495
λ
-
60
3600.00013892748
3600.00027802883
3600.00041738205
λ
60
3600.00013892749
3600.00027802885
3600.00041738205
λ
-
70
4900.00010206165
4900.00020421710
4900.00030650836
λ
70
4900.00010206165
4900.00020421711
4900.00030650836
λ
-
80
6400.00007813720
6400.00015632939
6400.00023460110
λ
80
6400.00007813721
6400.00015632940
6400.00023460110
λ
-
90
8100.00006173602
8100.00012350634
8100.00018532630
λ
90
8100.00006173602
8100.00012350635
8100.00018532634
λ
-
100
10000.00005000500
10000.00010003250
10000.00015009260
λ
100
10000.00005000500
10000.00010003260
10000.00015009260The estimations of the lengthsγ1,γ2,…,γ100 of the gaps are given in Table 5. It is known that [12] for large n the behavior of γn is sensitive to smoothness properties of the potential q. If q is m times differentiable, then γn=O(n-m). If q is analytic function, then γn=O(e-an) for some positive a. For the Mathieu operator T(p1) the following asymptotic formula holds: γn=O(4n/((n-1)!)2). Thus for large n the length γn of the nth gap is a very small number. Table 5 confirms this result for large n (see γn for n≥10). Moreover Table 5 shows that these results continue to hold for n>5. Since for the small values of n (n≤5) 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 n the theoretical results give the estimations of γn with very high accuracy.Table 5
Approximation of the lengths of the gaps.
p
1
p
2
p
3
γ
1
1.96935688950626
1.48577605581811
1.41120927072708
γ
2
0.45427620973738
2.29097727558352
1.90391794124493
γ
3
0.03062958739405
0.49917420034206
2.38589139848613
γ
4
0.00086225895372
0.15815420244566
0.47434407505152
γ
5
1.35221586674561 × 10−5
0.01489830930653
0.19405813943552
γ
6
1.35412271617952 × 10−7
0.00252680933036
0.07332561525572
γ
7
9.41085431804822 × 10−10
0.00015065970523
0.00830010463081
γ
8
1.09992015495664 × 10−11
1.76703419043633 × 10−5
0.00229625392370
γ
9
5.82645043323282 × 10−13
7.72435271301219 × 10−7
0.00053092953526
γ
10
1.22213350550737 × 10−12
6.92769646093439 × 10−8
4.64903052659338 × 10−5
γ
20
5.11590769747272 × 10−12
4.88853402202949 × 10−12
9.09494701772928 × 10−13
γ
30
8.64019966684282 × 10−12
4.54747350886464 × 10−13
4.43378667114303 × 10−12
γ
40
3.41060513164848 × 10−12
3.63797880709171 × 10−12
1.02318153949454 × 10−11
γ
50
3.18323145620525 × 10−12
6.82121026329696 × 10−12
5.45696821063757 × 10−12
γ
60
7.27595761418343 × 10−12
1.90993887372315 × 10−11
4.09272615797818 × 10−12
γ
70
3.63797880709171 × 10−12
5.45696821063757 × 10−12
2.72848410531878 × 10−12
γ
80
6.3664629124105 × 10−12
5.45696821063757 × 10−12
1.81898940354586 × 10−12
γ
90
1.81898940354586 × 10−12
6.3664629124105 × 10−12
4.09272615797818 × 10−11
γ
100
1.09139364212751 × 10−11
2.91038304567337 × 10−11
0It is natural and well known that for large eigenvalues the asymptotic method gives us approximations with smaller errors. Since the method of Section3 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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