Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order

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Volume 2010, Article ID 720235,21pages doi:10.1155/2010/720235

Research Article

Accurate Asymptotic Formulas for

Eigenvalues and Eigenfunctions of

a Boundary-Value Problem of Fourth Order

Hamza Menken

Mathematics Department, Science and Arts Faculty, Mersin University, 33343 Mersin, Turkey Correspondence should be addressed to Hamza Menken,[email protected]

Received 7 July 2010; Accepted 9 November 2010

Academic Editor: I. T. Kiguradze

Copyrightq2010 Hamza Menken. 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.

In the present paper, we consider a nonself-adjoint fourth-order diﬀerential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions.

1. Introduction

In the present work, we consider a nonself-adjoint fourth-order operator which is generated by the periodic boundary conditions:

y4_q_x_y_λy, ₀_≤_x_≤_π_, _1.1

yj0−yjπ 0, j0,1,2,3, 1.2

where qx is a complex-valued function. Without lose of generality, we can assume that _π

0 qxdx0.

Spectral properties of Sturm-Liouville operator which is generated by the periodic and antiperiodic boundary conditions have been investigated by many authors, the results on this direct and references are given details in the monographs1–5.

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using the obtained asymptotic formulas for eigenvalues and eigenfunctions, the basis properties of the root functions of the operators were investigated.

The paper is organized as follows. In Section 2, we compute new asymptotic expression of the fundamental solutions of 1.1. In Section 3, we obtain new accurate asymptotic estimates for the eigenvalues. In Section 4, we have asymptotic formulas for eigenfunctions under the distinct conditions onqx.

2. The Expression of the Fundamental Solutions

It is well known thatsee2, page 92if the complexs-planes4 _λ_{is divided into eight}

sectorsS 0,7, defined by the inequalities

π

8 ≤args≤

1π

8 ,

0,7, 2.1

then in each of these sectors1.1has four linear independent solutionsykx, s k1,2,3,4,

which are regular with respect tosin the sectorSlfor|s|suﬃciently large and which satisfy

the relation

ykx, s eωksx

₇

υ0

uv,kx

sv O

1

s8 , k1,2,3,4, 2.2

where the numbersωk are the fourth roots of unity, that is,ω1 −ω4 iandω2 −ω3 1.

In general, the term Os−N1_{at the formula} _2.2_{depends upon the smoothness of the}

function qx. If qx has mcontinuous derivatives, then one can assert the existence of a representation2.2withN m3. Here, we assume thatqx ∈C4₀_{, π}_{. The functions}

uυ,kxsatisfy the following recursion relations:

4u_υ,kx 6ω3_ku_υ₋₁_,kx 4ω_k2u_υ₋₂_,kx

ωku_υ4₋₃_,kx ωkqxuυ−3,kx 0, uυ,kx≡0, υ <0.

2.3

Let us put, moreover, u0,kx ≡ 1, uυ,k0 ≡ 0, for υ ≥ 1. Thus, the functions uυ,kxare

uniquely determined. Thus, we can find from2.3that

u0,kx 1, u1,kx 0, u2,kx 0, u3,kx −

1 4ω3

k

_x

0

qtdt,

u4,kx

3 8

qx−q0, u5,kx −

5ω3

k

16

qx−q0,

u6,kx

3ω2

k

32

qx−q0ω

2

k

32 x

0

qtdt 2

,

u7,kx −ωk

64

qx−q0−3ωk

32

qx−q0 x

0

qtdt− 3ωk

32 _x

0

q2_t_dt.

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3. The Asymptotic Formulas of Eigenvalues

It follows from the classical investigations see 4, page 65 that the eigenvalues of the problem1.1,1.2 in0,1 consist of the pairs of the sequences{λk,1},{λk,2} satisfying

the following asymptotic formula:

λk,1λk,2O

k1/2 2kπ4

1ξ0

k O

1

k3/2

3.1

for suﬃciently large integerk, whereξ0is a constant.

Theorem 3.1. Assume thatqx∈C4₀_{, π}_{. Then, the eigenvalues of the boundary-value problem}

1.1,1.2form two infinite sequencesλk,1, λk,2 kN,N1, . . ., whereNis a big positive integer

and have the following asymptotic formulas:

λk,1 2ki4

3 8π

_π

0 q2tdt

2ki4 O

1

k8

,

λk,2 2k4

3 8π

π

0 q2tdt

2k4 O

1

k8

.

3.2

Proof. By derivation of2.2up to third order with respect tox, the following relations are obtained:

y_kmx, s ωksmeωksx

⎡ ⎣7

υ0

u_υ,kmx

sυ O

1

s8

⎤

⎦_, _3.3

wherek1,2,3,4,m1,2,3 and

u₀m_,kx 1, u₁m_,kx 0, u₂m_,kx 0, u₃m_,kx − 1

4ω3

k

x

0

qtdt,

u₄1_,kx 1

8qx− 3

8q0, u

2

4,kx −

1 8qx−

3

8q0, u

3

4,kx −

3 8qx−

3 8q0,

u₅1_,kx ω

3

k

16q

_x 5ω3k

16 q

₀_, _u2

5,kx

3ω3_k

16 q

_x 5ω3k

16 q ₀_,

u₅3_,kx ω

3

k

16q

_x 5ωk3

(4)

u₆1_,kx −5ω

2

k

32

qx q0ω

2

k

32

x

0

qtdt 2

,

u₆2_,kx −3ω

2

k

32 q

_x₋5ωk2

32 q

₀ ω2k

32 x

0

qtdt 2

,

u₆3_,kx 3ω

2

k

32 q

_x₋5ω2k

32 q

₀ωk2

32

x

0

qtdt 2

,

u₇1_,kx 9ωk

64 q

_x ωk

64q

₀₋ωk

32

qx−3q0 x

0

qtdt−3ωk

32 _x

0

q2tdt,

u₇2_,kx −ωk

64q

_x ωk

64q

₀ ωk

32

qx 3q0 x

0

qtdt−3ωk

32 _x

0

q2tdt,

u₇3_,kx −7ωk

64 q

_x ωk

64q

₀ 3ωk

32

qx 3q0 x

0

qtdt− 3ωk

32 _x

0

q2_t_dt.

3.4

Now let us substitute all these expressions into the characteristic determinant

Δs

U1

y1

U1

y2

U1

y3

U1

y4

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

U4

y1

U4

y2

U4

y3

U4

y4

, 3.5

where

Uj1

yk

y_kjπ−y_kj0, j0,1,2,3. 3.6

By long computations, for suﬃciently large|s|, we obtain that

Δs 16is6

esπ

1−3 2

q0

s4 −

3 32

_π

0 q2tdt

s7 O

1

s8

−2

1−3

2

q0

s4 O

1

s8

e−sπ

1−3

2

q0

s4

3 32

_π

0 q2tdt

s7 O

1

s8

×

eisπ

1− 3i

32 π

0 q2tdt

s7 O

1

s8 −2

1O

1

s8

e−isπ

1 3i 32

_π

0 q2tdt

s7 O

1

s8 .

(5)

Multiplying the last equation by

esπ

13 2

q0

s4

3 32

_π

0 q2tdt

s7 O

1

s8 e

isπ

1 3i

32 _π

0 q2tdt

s7 O

1

s8 , 3.8

it becomes

esπ−

1 3

32 π

0 q2tdt

s7 O

1

s8

2

eisπ−

1 3i 32

π

0 q2tdt

s7 O

1

s8 2

. 3.9

Hence, byΔs 0, for suﬃciently large|s|, the following equations hold:

esπ−1 3 32

_π

0 q2tdt

s7 O

1

s8

, 3.10

eisπ−1 3i 32

_π

0 q2tdt

s7 O

1

s8

. 3.11

By Rouche’s theorem, we have asymptotic estimates for the rootssk,1 andsk,2,k N,N

1, . . . ,of3.10and3.11, respectively, whereNis a big positive integer

sk,12ki

3 32π

π

0 q2tdt

2ki7 O

1

k8

, 3.12

sk,22k 3

32π _π

0 q2tdt

2k7 O

1

k8

. 3.13

From the relations3.12,3.13and the relationsλk,j s4_k,j,j1,2, the asymptotic formulas

3.2are valid fork≥N.

4. The Asymptotic Formulas for the Eigenfunctions

Now, we obtain asymptotic formulas for eigenfunctions under the distinct conditions onqx. Case 1. Assume thatqx ∈C1₀_{, π}_{and the condition}_q_π₋_q₀_/_{0 holds. Based on the}

asymptotic expressions of the fundamental solutions of1.1and the asymptotic formulas for eigenvalues of the boundary-value problem1.1,1.2up to orderOs−5_{, the following}

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Theorem 4.1. If the condition qπ −q0/0 holds, then eigenfunctions of the boundary-value problem1.1,1.2corresponding the eigenvaluesλk,1andλk,2are of the form

yk,1x sin2kx−sinh2kx O

1

k

, 4.1

yk,2x cos2kx cosh2kx O

1

k

, 4.2

wherekis suﬃciently large integer.

Proof. Let us calculateUj1yυx, sk,1 j0,1,2up to orderOs−_k,5₁. Since

esk,1π−₁ 3 32

_π

0 q2tdt

s7

k,1

O

1

s8

k,1

, 4.3

we obtain that

U1

yυx, sk,1

3

8

qπ−q0 s4

k,1

O

1

s5

k,1

,

U2

yυx, sk,1

ωυsk,1

1 8

qπ−q0 s4_k,₁ O

1

s5_k,₁ ,

U3

yυx, sk,1

ωυsk,12

−1

8

qπ−q0 s4

k,1

O

1

s5

k,1

.

4.4

Follows from the conditionqπ−q0/0 thatUj1yυx, sk,1/0 for j 0,1,2. Thus, we

can seek eigenfunctionyk,1xcorrespondingλk,1in the form

yk,1x

y1x, sk,1 y2x, sk,1 y3x, sk,1 y4x, sk,1

U1

y1

U1

y2

U1

y3

U1

y4

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

. 4.5

Then,

yk,1x y1x, sk,1

U1

y2

U1

y3

U1

y4

U2

y2

U2

y3

U2

y4

U3

y2

U3

y3

U3

y4

−y2x, sk,1

U1

y1

U1

y3

U1

y4

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

(7)

y3x, sk,1 U1 y1 U1 y2 U1 y4 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4

−y4x, sk,1

U1 y1 U1 y2 U1 y3 U2 y1 U2 y2 U2 y3 U3 y1 U3 y2 U3 y3 . 4.6

By simple computations, we obtain U1 y2 U1 y3 U1 y4 U2 y2 U2 y3 U3 y4 U3 y2 U3 y3 U3 y4 − 3 27

qπ−q03 s9_k,₁

1O

1

sk,1

, U1 y1 U1 y3 U1 y4 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 − 3i 27

qπ−q03 s9

k,1

1O

1

sk,1

qπ−q03 s9_k,₁

1O

1

sk,1

, U1 y1 U1 y2 U1 y3 U2 y1 U2 y2 U2 y3 U3 y1 U3 y2 U3 y3 − 3 27

qπ−q03 s9

k,1

1O

1

sk,1

.

4.7

Hence, using the formula2.2, we can write

yk,1x −3

27

qπ−q03 s9_k,₁

eisk,1x−_iesk,1x_ie−sk,1x−_e−isk,1x_O

1

sk,1

−3

27

qπ−q03 s9

k,1

2isinsk,1x−2isinhsk,1xO

1

sk,1

−3i

26

qπ−q03 s9_k,₁

sinsk,1x−sinhsk,1xO

1

sk,1

.

4.8

Therefore, for the normalized eigenfunction, we get

yk,1x sinsk,1x−sinhsk,1xO

1

sk,1

(8)

Using the relations3.3and3.12, for suﬃciently large integerk, we obtain4.1

1

k

. 4.10

Similarly, since Uj1yυx, sk,1/0 for j 1,2,3, we can seek eigenfunction yk,2x

correspondingλk,2in the form

yk,2x

U2 y1 U2 y2 U2 y3 U2 y4 U3 y1 U3 y2 U3 y3 U3 y4 U4 y1 U4 y2 U4 y3 U4 y4 . 4.11 Then,

yk,2x y1x, sk,2

U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4

−y2x, sk,2

y3x, sk,2

−y4x, sk,2

By similar computations we obtain U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 3i 27

qπ−q03 s6

k,2

1O

1

sk,2

qπ−q03 s6_k,₂

1O

1

sk,2

(9)

U2

y1

U2

y2

U2

y4

U3

y1

U3

y2

U3

y4

U4

y1

U4

y2

U4

y4

3i

27

qπ−q03 s6

k,2

1O

1

sk,2

,

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

U4

y1

U4

y2

U4

y3

−

3i

27

qπ−q03 s6_k,₂

1O

1

sk,2

.

4.13

yk,2x

3i

27

qπ−q03 s6_k,₂

eisk,2x_esk,2x_e−sk,2x_e−isk,2x_O

1

sk,2

3i 27

qπ−q03 s6_k,₂

2 cossk,2x2 coshsk,2xO

1

sk,2

3i 26

qπ−q03 s6

k,1

cossk,2xcoshsk,2xO

1

sk,2

.

4.14

yk,2x cossk,2xcoshsk,2xO

1

sk,2

. 4.15

Hence, for suﬃciently large integerk, we obtain4.2

1

k

. 4.16

Case 2. Assume thatqx∈C20, πand the conditionsqπ−q0 0 andqπ−q0/0 hold. Based on the asymptotic expressions of the fundamental solutions of 1.1 and the asymptotic formulas for eigenvalues of the boundary-value problem1.1,1.2up to order

Os−6_{, the following result is valid.}

Theorem 4.2. If the conditionsqπ−q0 0andqπ−q0/0hold, then eigenfunctions of the boundary-value problem1.1,1.2corresponding the eigenvaluesλk,1andλk,2are of the form

yk,1x cos2kx−cosh2kx O

1

k

, 4.17

1

k

, 4.18

(10)

Proof. It is clear that

U1

yυx, sk,1

3

8

qπ−q0 s4

k,1

5ωυ

16

qπ−q0 s5

k,1

O

1

s6

k,1

,

U2

yυx, sk,1

ωυsk,1

1 8

qπ−q0 s4_k,₁ −

ωυ

16

qπ−q0 s5_k,₁

O

1

s6

k,1

,

U3

yυx, sk,1

ωυsk,12

−1

8

qπ−q0 s4

k,1

−3ωυ

16

qπ−q0 s5_k,₁

O

1

s6_k,₁ .

4.19

It follows from the conditionsqπ−q0 0,qπ−q0/0 thatUj1yυx, sk,1/0 for

j0,1,2. Thus, we can seek eigenfunctionyk,1xcorrespondingλk,1in the form

yk,1x

U1

y1

U1

y2

U1

y3

U1

y4

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

. 4.20

Then,

yk,1x y1x, sk,1

U1

y2

U1

y3

U1

y4

U2

y2

U2

y3

U2

y4

U3

y2

U3

y3

U3

y4

−y2x, sk,1

U1

y1

U1

y3

U1

y4

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

(11)

y3x, sk,1 U1 y1 U1 y2 U1 y4 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4

−y4x, sk,1

By simple computations, we have U1 y2 U1 y3 U1 y4 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 15i 210

qπ−q03 s12_k,₁

1O

1

sk,1

, U1 y1 U1 y3 U1 y4 U2 y1 U3 y3 U2 y4 U3 y1 U3 y3 U3 y4 15i 210

qπ−q03 s12

k,1

1O

1

sk,1

qπ−q03 s12_k,₁

1O

1

sk,1

qπ−q03 s12

k,1

1O

1

sk,1

.

4.22

yk,1x

15i

210

qπ−q03 s12

k,1

eisk,1x−_esk,1x−_e−sk,1x_e−isk,1x_O

1

sk,1

15i 29

qπ−q03 s12_k,₁

cossk,1x−coshsk,1xO

1

sk,1

.

4.23

yk,1x cossk,1x−coshsk,1xO

1

sk,1

(12)

Using the relations3.3and3.12, for suﬃciently large integerk, we obtain4.17:

1

k

. 4.25

In similar way, we can seek eigenfunctionyk,2xcorrespondingλk,2in the form

yk,2x

U2 y1 U2 y2 U2 y3 U2 y4 U3 y1 U3 y2 U3 y3 U3 y4 U4 y1 U4 y2 U4 y3 U4 y4 . 4.26 Then,

yk,2x y1x, sk,2

−y2x, sk,2

y3x, sk,2

−y4x, sk,2

By simple computations, we get U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 3 210

qπ−q03 s9

k,2

1O

1

sk,2

qπ−q03 s9_k,₂

1O

1

sk,2

(13)

U2

y1

U2

y2

U2

y4

U3

y1

U3

y2

U3

y4

U4

y1

U4

y2

U4

y4

3i

210

qπ−q03 s9

k,2

1O

1

sk,2

,

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

U4

y1

U4

y2

U4

y3

3 210

qπ−q03 s9_k,₂

1O

1

sk,2

.

4.28

yk,2x

3 210

qπ−q03 s9_k,₂

eisk,2x−_iesk,2x_ie−sk,2x−_e−isk,2x_O

1

sk,2

3i 29

qπ−q03 s9

k,2

sinsk,2x−sinhsk,2xO

1

sk,2

.

4.29

yk,2x sinsk,2x−sinhsk,2xO

1

sk,2

O

1

sk,2

. 4.30

Hence, for suﬃciently large integerk, we obtain4.18:

1

k

. 4.31

Case 3. Assume thatqx ∈ C3₀_{, π}_{and the conditions}_qj_π₋_qj₀ ₀_{, j} ₀_,_{1 and} qπ−q0/0 hold. Based on the asymptotic expressions of the fundamental solutions of 1.1and the asymptotic formulas for eigenvalues of the boundary-value problem1.1,1.2 up to orderOs−7_{, the following result is valid.}

Theorem 4.3. If the conditions qj_π₋_qj₀ ₀_{, j} ₀_,₁_and _q_π₋_q₀_/₀ _{hold, then} eigenfunctions of the boundary-value problem1.1,1.2corresponding the eigenvaluesλk,1andλk,2

are of the form

yk,1x sin2kx sinh2kx O

1

k

, 4.32

1

k

4.33

(14)

U1

yυx, sk,1

3

8

qπ−q0 s4

k,1

5ωυ

16

qπ−q0 s5

k,1

−5ω2υ

32

qπ−q0 s6

k,1

O

1

s7

k,1

,

U2

yυx, sk,1

ωυsk,1

1 8

qπ−q0 s4_k,₁ −

ωυ

16

qπ−q0 s5_k,₁ −

5ω2

υ

32

qπ−q0 s6_k,₁

O

1

s7_k,₁

,

U3

yυx, sk,1

ωυsk,12

−1

8

qπ−q0 s4

k,1

−3ωυ

16

qπ−q0 s5

k,1

− 3ωυ2

32

qπ−q0 s6

k,1

O

1

s7

k,1

.

4.34

From the conditions qj_π ₋ _qj₀ ₀_j ₀_,₁ _and _q_π ₋ _q₀_/_{0, we have} Uj1yυx, sk,1/0 forj 0,1,2. Thus, we can seek eigenfunctionyk,1xcorrespondingλk,1

in the form

yk,1x

U1

y1

U1

y2

U1

y3

U1

y4

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

. 4.35

Then,

yk,1x y1x, sk,1

U1

y2

U1

y3

U1

y4

U2

y2

U2

y3

U2

y4

U3

y2

U3

y3

U3

y4

−y2x, sk,1

U1

y1

U1

y3

U1

y4

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

y3x, sk,1

U1

y1

U1

y2

U1

y4

U2

y1

U2

y2

U2

y4

U3

y1

U3

y2

U3

y4

−y4x, sk,1

U1

y1

U1

y2

U1

y3

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

.

(15)

By simple calculations, we get

U1

y2

U1

y3

U1

y4

U2

y2

U2

y3

U2

y4

U3

y2

U3

y3

U3

y4

−

75 213

qπ−q03 s15

k,1

1O

1

sk,1

,

U1

y1

U1

y3

U1

y4

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

75i

213

qπ−q03 s15_k,₁

1O

1

sk,1

,

U1

y1

U1

y2

U1

y4

U2

y1

U2

y2

U2

y4

U3

y1

U3

y2

U3

y4

75i

213

qπ−q03 s15_k,₁

1O

1

sk,1

,

U1

y1

U1

y2

U1

y3

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

−

75 213

qπ−q03 s15

k,1

1O

1

sk,1

.

4.37

yk,1x −

75 213

qπ−q03 s15_k,₁

eisk,1x_iesk,1x−_ie−sk,1x−_e−isk,1x_O

1

sk,1

−75i

212

qπ−q03 s15

k,1

sinsk,1xsinhsk,1xO

1

sk,1

.

4.38

yk,1x sinsk,1xsinhsk,1xO

1

sk,1

. 4.39

Using the relations3.3and3.12, for suﬃciently large integerk, we obtain4.32

1

k

. 4.40

yk,2x

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

U4

y1

U4

y2

U4

y3

U4

y4

(16)

Then,

yk,2x y1x, sk,2

−y2x, sk,2

y3x, sk,2

−y4x, sk,2

By simple computations, we get U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 − 45i 213

qπ−q03 s12

k,2

1O

1

sk,2

qπ−q03 s12

k,2

1O

1

sk,2

qπ−q03 s12_k,₂

1O

1

sk,2

qπ−q03 s12

k,2

1O

1

sk,2

.

4.43

By the formula2.2, we can write

yk,2x −

45i

213

qπ−q03 s12

k,2

eisk,2x−_esk,2x−_e−sk,2x_e−isk,2x_O

1

sk,2

45i 212

qπ−q03 s12_k,₂

cossk,2x−coshsk,2xO

1

sk,2

.

(17)

yk,2x cossk,2x−coshsk,2xO

1

sk,2

. 4.45

Hence, for suﬃciently large integerk, we obtain the relation4.33

1

k

. 4.46

Case 4. Assume thatqx ∈ C4₀_{, π}_{and the conditions} _qj_π₋_qj₀ _0,_j ₀_,_{2 and}

qπ−q0/0 hold. Based on the asymptotic expressions of the fundamental solutions of 1.1and the asymptotic formulas for eigenvalues of the boundary-value problem1.1,1.2 up to orderOs−8_{, the following result is valid.}

Theorem 4.4. If the conditionsqjπ−qj0 0, j 0,2 and qπ−q0/0 hold, then eigenfunctions of the boundary-value problem1.1,1.2corresponding the eigenvaluesλk,1andλk,2

are of the form

1

k

, 4.47

1

k

, 4.48

wherekis suﬃciently large integer.

U1

yυx, sk,1

3

8

qπ−q0 s4_k,₁

5ωυ

16

qπ−q0 s5_k,₁

−5ω2υ

32

qπ−q0 s6

k,1

−ωυ

64

qπ−q0 s7

k,1

O

1

s8

k,1

,

U2

yυx, sk,1

ωυsk,1

1 8

qπ−q0 s4

k,1

−ωυ

16

qπ−q0 s5_k,₁

− 5ω2υ

32

qπ−q0 s6_k,₁

9ωυ

64

qπ−q0 s7

k,1

O

1

s8_k,₁

(18)

U3

yυx, sk,1

ωυsk,12

−1

8

qπ−q0 s4

k,1

−3ωυ

16

qπ−q0 s5

k,1

− 3ω2υ

32

qπ−q0 s6

k,1

−ωυ

64

qπ−q0 s7 k,1 O 1 s8 k,1 . 4.49

From the conditions qj_π ₋ _qj₀ ₀ _j ₀_,₂ _and _q_π ₋ _q₀_/_{0, we have} Uj1yυx, sk,1/0 forj 0,1,2. Thus, we can seek eigenfunctionyk,1xcorrespondingλk,1

in the form

yk,1x

U1 y1 U1 y2 U1 y3 U1 y4 U2 y1 U2 y2 U2 y3 U2 y4 U3 y1 U3 y2 U3 y3 U3 y4 . 4.50 Then

yk,1x y1x, sk,1

−y2x, sk,1

y3x, sk,1

−y4x, sk,1

By simple computations, we get U1 y2 U1 y3 U1 y4 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 9i 216

qπ−q03 s18

k,1

1O

1

sk,1

qπ−q03 s18

k,1

1O

1

sk,1

(19)

U1

y1

U1

y3

U1

y4

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

9i

216

qπ−q03 s18

k,1

1O

1

sk,1

,

U1

y1

U1

y2

U1

y3

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

−

9i

216

qπ−q03 s18_k,₁

1O

1

sk,1

.

4.52

yk,1x

9i

216

qπ−q03 s18_k,₁

eisk,1x_esk,1x_e−sk,1x_e−isk,1x_O

1

sk,1

9i 215

qπ−q03 s18

k,1

cossk,1xcoshsk,1xO

1

sk,1

.

4.53

yk,1x cossk,1xcoshsk,1xO

1

sk,1

. 4.54

Using the relations3.3and3.12, for suﬃciently large integerk, we obtain4.47:

1

k

. 4.55

yk,2x

U2

y1

U2

y2

U2

y3

U2

y4

U3

y1

U3

y2

U3

y3

U3

y4

U4

y1

U4

y2

U4

y3

U4

y4

(20)

By simple computations, we get

U2

y2

U2

y3

U2

y4

U3

y2

U3

y3

U3

y4

U4

y2

U4

y3

U4

y4

−

63 216

qπ−q03 s15

k,2

1O

1

sk,2

,

U2

y1

U2

y3

U2

y4

U3

y1

U3

y3

U3

y4

U4

y1

U4

y3

U4

y4

63i

216

qπ−q03 s15_k,₂

1O

1

sk,2

,

U2

y1

U2

y2

U2

y4

U3

y1

U3

y2

U3

y4

U4

y1

U4

y2

U4

y4

63i

216

qπ−q03 s15_k,₂

1O

1

sk,2

,

U2

y1

U2

y2

U2

y3

U3

y1

U3

y2

U3

y3

U4

y1

U4

y2

U4

y3

−

63 216

qπ−q03 s15

k,2

1O

1

sk,2

.

4.57

yk,2x −

63 216

qπ−q03 s15_k,₂

eisk,2x_iesk,2x−_ie−sk,2x−_e−isk,2x_O

1

sk,2

−63i

215

qπ−q03 s15

k,2

sinsk,2xsinhsk,2xO

1

sk,2

.

4.58

yk,2x sinsk,2xsinhsk,2xO

1

sk,2

. 4.59

Hence, for suﬃciently large integerk, we obtain the relation4.48

1

k

. 4.60

Acknowledgments

(21)

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