# Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

Authors: Manfred Möller; Bertin Zinsou
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameterλ and which has separable boundary conditions depending linearly on λ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.


---

## Body

## 1. Introduction

Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathematical challenges and their applications in physics and engineering. Classical Sturm-Liouville problems have been extended to higher-order differential equations and to differential equations with eigenvalue parameter dependent boundary conditions. For example, the generalized Regge problem is realised by a second order differential operator which depends quadratically on the eigenvalue parameter and which has eigenvalue parameter dependent boundary conditions, see [1]. The particular feature of this problem is that the coefficient operators of this pencil are self-adjoint, and it is shown in [1] that this gives some a priori knowledge about the location of the spectrum. In [2], this approach has been extended to a fourth order differential equation describing small transversal vibrations of a homogeneous beam compressed or stretched by a force g. Again, this problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coefficient operators are self-adjoint. In [3], we have considered this fourth order differential equation with general two-point boundary conditions which depend linearly on the eigenvalue parameter. Necessary and sufficient conditions such that the associated operator pencil consists of self-adjoint operators have been obtained. In [4, 5], we have derived eigenvalue asymptotics associated with particular boundary conditions. In this paper, we are considering the case of separable boundary conditions where all four of these boundary conditions depend on the eigenvalue parameter.Other recent results on fourth order differential operators whose boundary conditions depend on the eigenvalue parameter but which are represented by linear operator pencils, include spectral asymptotics and basis properties, see [6–8].In Section2, we introduce the operator pencil associated with the eigenvalue problem (1), (2), and we derive the boundary conditions such that the operators in the pencil are self-adjoint. In Section 3, we obtain the location of the spectrum and the asymptotic distribution of the eigenvalues for the case g=0. In Section 4, we prove that the boundary value problem under investigation is Birkhoff regular, which implies that the eigenvalues for general g are small perturbations of the eigenvalues for g=0. Hence, in Section 5, we derive the first four terms of the asymptotics of the eigenvalues and compare them to those obtained for the boundary conditions considered in [5].


## 2. The Quadratic Operator PencilL

On the interval[0,a], we consider the boundary value problem
(1)y(4)-(gy′)′=λ2y,(2)Bj(λ)y=0,j=1,2,3,4,
where g∈C1[0,a] is a real valued function and (2) is separated boundary conditions with the Bj depending linearly on the eigenvalue parameter λ. The boundary conditions (2) are taken at the endpoint 0 for j=1,2 and at the endpoint a>0 for j=3,4. Further, we assume for simplicity that
(3)Bj(λ)y=y[pj](aj)+iαεjλy[qj](aj),
where aj=0 for j=1,2 and aj=a for j=3,4, 0≤qj<pj≤3, α>0, and εj∈ℂ∖{0}. We recall that the quasi-derivatives associated with (1) are given by
(4)y[0]=y,y[1]=y′,y[2]=y′′,y[3]=y(3)-gy′,y[4]=y(4)-(gy′)′,
see [9, page 26]. In order to have independent boundary conditions, we will also assume that the numbers p1, q1, p2, q2 as well as the numbers p3, q3, p4, q4 are mutually disjoint.Recall that in applications, using separation of variables, the parameterλ emanates from derivatives with respect to the time variable in the original partial differential equation, and it is reasonable that the highest space derivative occurs in the term without time derivative. Thus, the most relevant boundary conditions would have qj<pj for j=1,…,4.Further assumptions on thepj, qj, and εj will be made later and will be justified by the requirements on the operator pencil which we are going to define now.For rather generic boundary conditions, a quadratic operator pencil has been associated in [3], and we will now recall notations and results from [3] which are relevant in our case.We denote byU the collection of the boundary conditions (2) and define the following operators related to U:
(5)U0y=(y[pj](aj))j=14,U1y=(εjy[qj](aj))j=14,-----------------------y∈W42(0,a),
where W42(0,a) is the Sobolev space of order 4 on the interval (0,a). We consider the linear operators A(U), K, and M in the space L2(0,a)⊕ℂ4 with domains
(6)𝒟(A(U))={y~=(yU1y):y∈W42(0,a)},𝒟(K)=𝒟(M)=L2(0,a)⊕ℂ4,
given by
(7)(A(U))y~=(y[4]U0y)fory~∈𝒟(A(U)),K=(000I),M=(I000).
It is easy to check that K≥0, M≥0, M+K=I, and M|𝒟(A(U))>0. We associate a quadratic operator pencil
(8)L(λ,α)=λ2M-iαλK-A(U),λ∈ℂ,
in the space L2(0,a)⊕ℂ4 with the problem (1), (2). We observe that (8) is an operator representation of the eigenvalue problem (1), (2) in the sense that a function y satisfies (1), (2) if and only if L(λ,α)y~=0.For the boundary conditions (2) with the assumptions made so far, [3, Theorem 1.2] leads to the following.Proposition 1.
The differential operatorA(U) associated with (1), (2) is self-adjoint if and only if for j=1,…,4; the numbers pj, qj, and εj satisfy the following conditions: pj+qj=3, εj=1 if qj is even in case aj=0 or odd in case aj=a, and εj=-1, otherwise.Since we aim to consider the boundary eigenvalue problem (1), (2) in case that A(U) is self-adjoint, the boundary operators are, up to permutation,
(9)B1(λ)y=y′′(0)-iαλy′(0),(10)B2(λ)y=y[3](0)+iαλy(0),(11)B3(λ)y=y′′(a)+iαλy′(a),(12)B4(λ)y=y[3](a)-iαλy(a).
As in [4, Proposition 2.3], we obtain the following.Proposition 2.
The operator pencilL(·,α) is a Fredholm valued operator function with index 0. The spectrum of the Fredholm operator L(·,α) consists of discrete eigenvalues of finite multiplicities, and all eigenvalues of L(·,α), α≥0, lie in the closed upper half-plane and on the imaginary axis and are symmetric with respect to the imaginary axis.Proof.
As in [2, Section 3], we can argue that for all λ∈ℂ, L(λ,α) is a relatively compact perturbation of L(0,0), where L(0,0) is well known to be a Fredholm operator. The statement on the location of the spectrum follows as in [2, Lemma 3.1].


## 3. Asymptotics of Eigenvalues forg=0

In this section, we investigate the boundary value problem (1), (2) with g=0. We count all eigenvalues with their proper multiplicities and develop a formula for the asymptotic distribution of the eigenvalues, which we will use to obtain the corresponding formula for general g. Observe that for g=0, the quasi-derivatives y[j] coincide with the standard derivatives y(j). We take the canonical fundamental system yj(·,λ), j=1,…,4, of (1) with yj(m)(0)=δj,m+1 for m=0,…,3. It is well known that the functions yj(·,λ) are analytic on ℂ with respect to λ. Putting
(13)M(λ)=(Bi(λ)yj(·,λ))i,j=14,
the eigenvalues of the boundary value problem (1), (2) are the eigenvalues of the analytic matrix function M, where the corresponding geometric and algebraic multiplicities coincide, see [10, Theorem 6.3.2].Settingλ=μ2 and
(14)y(x,μ)=12μ3sinh(μx)-12μ3sin(μx),
it is easy to see that
(15)yj(x,λ)=y(4-j)(x,μ),j=1,…,4.The first and the second rows ofM(λ) have exactly two nonzero entries (for λ≠0), and these nonzero entries are: B1(λ)y2(·,λ)=-iαμ2 and B1(λ)y3(·,λ)=1 while B2(λ)y1(·,λ)=iαμ2 and B2(λ)y4(·,λ)=1. Therefore,
(16)M(λ)=(0-iαμ210iαμ2001B3(μ2)y1B3(μ2)y2B3(μ2)y3B3(μ2)y4B4(μ2)y1B4(μ2)y2B4(μ2)y3B4(μ2)y4).
An expansion of detM(λ)=ϕ(μ) gives
(17)ϕ(μ)=-α2μ4Φ1(μ)+iαμ2Φ2(μ)+iαμ2Φ3(μ)+Φ4(μ),
where
(18)Φj(μ)=det(B3(μ2)yσj,1(·,μ)B3(μ2)yσj,2(·,μ)B4(μ2)yσj,1(·,μ)B4(μ2)yσj,2(·,μ)),
with
(19)(σj,1,σj,2)={(3,4)ifj=1,(1,3)ifj=2,(2,4)ifj=3,(1,2)ifj=4.
In view of (11), (12), we get that
(20)Φj(μ)-=[iαμ2{yσj,1′(a,μ)yσj,2(3)(a,μ)---------yσj,2′(a,μ)yσj,1(3)(a,μ)--------+yσj,2′′(a,μ)yσj,1(a,μ)---------yσj,1′′(a,μ)yσj,2(a,μ)}---+α2μ4{yσj,1′(a,μ)yσj,2(a,μ)----------yσj,2′(a,μ)yσj,1(a,μ)}---+yσj,1′′(a,μ)yσj,2(3)(a,μ)----yσj,2′′(a,μ)yσj,1(3)(a,μ)].
Observing that the yj are given by (14) and (15) a straightforward calculation leads to
(21)Φ1(μ)=12iαμ(sin(μa)cosh(μa)+cos(μa)sinh(μa))-+12μiα(sin(μa)cosh(μa)--------cos(μa)sinh(μa))--12α2(1-cos(μa)cosh(μa))-+12(1+cos(μa)cosh(μa)),Φ2(μ)=-iαμ4sin(μa)sinh(μa)-+iαμ2cos(μa)cosh(μa)-12(1+α2)-×μ3(sin(μa)cosh(μa)+cos(μa)sinh(μa)),Φ3(μ)=iαμ2cos(μa)cosh(μa)+iαsin(μa)sinh(μa)--12(1+α2)-×μ(sin(μa)cosh(μa)-cos(μa)sinh(μa)),Φ4(μ)=-12iαμ5(sin(μa)cosh(μa)+cos(μa)sinh(μa))--12iαμ3(sin(μa)cosh(μa)-cos(μa)sinh(μa))--12α2μ4(1+cos(μa)cosh(μa))-+12μ4(1-cos(μa)cosh(μa)).
It follows that the term with the highest μ-power in ϕ comes from Φ2 and is a nonzero multiple of
(22)ϕ0(μ)≔μ6sin(μa)sinh(μa).The following result on the zeros ofϕ0, with proper counting, is obvious.Lemma 3.
ϕ
0 has a zero of multiplicity 8 at 0, simple zeros at
(23)μ~k=(k-2)πa,k=3,4,…,
simple zeros at -μ~k, μ~-k=iμ~k and -iμ~k for k=3,4…, and no other zeros.Proposition 4.
Forg=0, there exists a positive integer k0 such that the eigenvalues λ^k, k∈ℤ∖{0}, counted with multiplicity, of the problem (1), (9)–(12) can be enumerated in such a way that the eigenvalues λ^k are pure imaginary for |k|<k0, and λ^-k=-λ^k¯ for k≥k0. For k>0, we can write that λ^k=μ^k2, where the μ^k have the following asymptotic representation as k→∞:
(24)μ^k=(k-2)πa+o(1).
In particular, the number of pure imaginary eigenvalues is even.Proof.
It follows from (17) and (21) that
(25)ϕ(μ)=α2μ6sin(μa)sinh(μa)-i(α3+α)-×μ5(sin(μa)cosh(μa)+cos(μa)sinh(μa))--α2μ4(1+3cos(μa)cosh(μa))--α2μ2sin(μa)sinh(μa)-+12α4μ4(1-cos(μa)cosh(μa))-+12μ4(1-cos(μa)cosh(μa))-i(α3+α)-×μ3(sin(μa)cosh(μa)-cos(μa)sinh(μa)).
Up to the constant factor α2, the first term equals ϕ0(μ). Considering that, therefore,
(26)ϕ1(μ)≔ϕ(μ)-α2ϕ0(μ)ϕ0(μ)
for ϕ0(μ)≠0, that is, sin(μa)≠0 and sinh(μa)≠0. It follows that
(27)ϕ1(μ)=-i(α3+α)μ(cot(μa)+coth(μa))--α2μ2[1sin(μa)sinh(μa)+3cot(μa)coth(μa)]--α2μ4-+α42μ2[1sin(μa)sinh(μa)-cot(μa)coth(μa)]-+12μ2[1sin(μa)sinh(μa)-cot(μa)coth(μa)]--i(α3+α)μ3(cot(μa)-coth(μa)).
Fixε∈(0,π/2a), and for k=3,4,…, let Rk,ε be the squares determined by the vertices (k-2)(π/a)±ε±iε, k=3,4,…. These squares do not intersect due to ε<π/2a. There exists C1(ε)>0 such that |coth(μa)|<C1(ε) for all μ on the squares Rk,ε, k≥3. By periodicity, there are numbers C2(ε) and C3(ε) such that |sin(μa)|>C2(ε) and |cot(μa)|<C3(ε) for all μ on the boundary of the squares Rk,ε for all k≥3. Since |sinh(μa)|≥|sinh((Reμ)a)|, it follows that there is k1(ε)∈ℕ such that for all μ on the squares Rk,ε, where k>k1(ε), the estimate |ϕ1(μ)|<1 holds. By Lemma 3, ϕ0 has exactly one simple zero inside Rk,ε. Hence, it follows from Rouché’s theorem that there is exactly one (simple) zero μ^k of ϕ in each Rk,ε for k≥k1(ε), which proves the existence of zeros μ^k of ϕ with μ^k=μ~k+o(1) as k→∞.
Let(28)ϕ~0(μ)=α2μ6sin(μa)sinh(μa)--α2μ4(1+3cos(μa)cosh(μa))--α2μ2sin(μa)sinh(μa)-+12α4μ4(1-cos(μa)cosh(μa))-+12μ4(1-cos(μa)cosh(μa)),ϕ~1(μ)=-i(α3+α)-×μ5(sin(μa)cosh(μa)+cos(μa)sinh(μa))-i(α3+α)-×μ3(sin(μa)cosh(μa)-cos(μa)sinh(μa)).
Then, ϕ(μ)=ϕ~0(μ)+ϕ~1(μ), and for all μ∈ℂ,
(29)ϕ(iμ)=ϕ~0(μ)-ϕ~1(μ)=ϕ(μ¯)¯.
Observing that ϕ is an even function, it follows altogether that ϕ has zeros ±μ^k, ±μ^-k for k>k1(ε) with μ^-k=iμ^k¯.
We still have to show that the stated counting and asymptotic behaviour describes all zeros ofϕ. To this end, we are going to estimate ϕ1 on the squares Sk, k∈ℕ, whose vertices are ±(k+(1/2))(π/a)±i(k+(1/2))(π/a). For μ=x+iy, x, y∈ℝ and x≠0, we have
(30)coth(μa)=e(ax+iay)+e-(ax+iay)e(ax+iay)-e-(ax+iay)⟶±1
uniformly in y as x→±∞. Hence, there are ξ≥1 and k~0>0 such that for all k∈ℕ, k≥k~0 and γ∈ℝ,
(31)|coth(((k+(1/2))πa+iγ)a)|≤ξ.
Note that for k∈ℕ and γ∈ℝ,
(32)cot(((k+(1/2))πa+iγ)a)=-tan(iγa)=-itanh(γa).
Hence, we have shown that for k∈ℕ and γ∈ℝ,
(33)|cot(((k+(1/2))πa+iγ)a)|≤1.
Furthermore, we will make use of the estimates
(34)|sinh(((k+(1/2))πa+iγ)a)|≥|sinh((k+12)π)|,|sin(((k+(1/2))πa+iγ)a)|=cosh(γa)≥1,
which hold for all k∈ℤ and all γ∈ℝ. Therefore, it follows from (31), (33)-(34), and the symmetry of ϕ, see (29), that there is k~1≥k~0 such that |ϕ1(μ)|<1 for all μ∈Sk with k>k~1. Again from the definition of ϕ1 in (27) and Rouché’s theorem, we conclude that the functions ϕ0 and ϕ have the same number of zeros in the square Sk, for k∈ℕ with k≥k~1. Since ϕ0 has 4k+8 zeros inside Sk and thus 4k+8+4 zeros inside Sk+1, it follows that all such zeros of ϕ are accounted for by ±μ^j for 0<|j|≤k+2 and sufficiently large k.
Finally,λ^k=μ^k2 account for all eigenvalues of the problem (1)-(2) since each of these eigenvalues gives rise to two zeros of ϕ, counted with multiplicity. By Proposition 2, all eigenvalues with nonzero real part occur in pairs λ^k, -λ^k¯, which shows that we can index all such eigenvalues as λ^-k=-λ^k¯. Since there is an even number of remaining indices, the number of pure imaginary eigenvalues must be even.


## 4. Birkhoff Regularity

We refer to [10, Definition 7.3.1] for the definition of Birkhoff regularity.Proposition 5.
The boundary value problem (1), (9)–(12) is Birkhoff regular for α>0 with respect to the eigenvalue parameter μ given by λ=μ2.Proof.
The characteristic function of (1) as defined in [10, (7.1.4)] is π(ρ)=ρ4-1, and its zeros are ik-1, k=1,…,4. We can choose
(35)C(x,μ)=diag(1,μ,μ2,μ3)(i(k-1)(j-1))k,j=14
according to [10, Theorem 7.2.4.A]. The boundary condition (9)–(12) can be written in the form
(36)Bj(λ)y=B^j(μ)(y(aj),y′(aj),y′′(aj),y(3)(aj)),--------------------j=1,2,3,4.
Thus, the boundary matrices defined in [10, (7.3.1)] are given by
(37)W(0)(μ)=(B^1(μ)B^2(μ)00)C(0,μ),W(1)(μ)=(00B^3(μ)B^4(μ))C(a,μ),
where
(38)B^1(μ)=(0,-iαμ2,1,0),B^2(μ)=(iαμ2,-g(0),0,1),B^3(μ)=(0,iαμ2,1,0),B^4(μ)=(-iαμ2,-g(a),0,1).
Choosing C2(μ)=diag(μ3,μ3,μ3,μ3), it follows that C2(μ)-1W(j)(μ)=W0(j)+O(μ-1), where
(39)W0(0)=(-iααiα-α1-i-1i00000000),W0(1)=(00000000iα-α-iαα1-i-1i).
The Birkhoff matrices are
(40)W0(0)Δj+W0(1)(I-Δj),
where Δj, j=1,2,3,4 are the 4×4 diagonal matrices with 2 consecutive ones and 2 consecutive zeros in the diagonal in a cyclic arrangement, see [10, Definition 7.3.1 and Proposition 4.1.7]. It is easy to see that after a permutation of columns, the matrices (40) are block diagonal matrices consisting of 2×2 blocks taken from two consecutive columns (in the sense of cyclic arrangement) of the first two rows of W0(0) and the last two rows of W0(1), respectively. Hence, the determinants of the Birkhoff matrices (40) are
(41)(-1)j-1|-ijα-ij+1α(-i)j-1(-i)j||ij+2αij+3α(-i)j+1(-i)j+2|=(-1)j-1(-2α)(2α)=(-1)j4α2≠0.
Thus, the problem (1), (9)–(12) is Birkhoff regular for α>0.


## 5. Asymptotic Expansions of Eigenvalues

Withλ=μ2, D(μ)=det(Bi(μ2)yj(·,μ))i,j=14 defines a characteristic determinant of the problem (1), (9)–(12) with respect to the fundamental system yj, j=1,2,3,4 considered in Section 3. Observe that ϕ is the corresponding characteristic determinant for g=0. Due to the Birkhoff regularity, g only influences lower order terms in D. Together with the estimates in Section 3, it can be inferred that outside the interior of the small squares Rk,ε, -Rk,ε, iRk,ε, -iRk,ε around the zeros of ϕ0, |D(μ)-ϕ0(μ)|<|ϕ0(μ)| if |μ| is sufficiently large. Since the fundamental system yj, j=1,2,3,4, depends analytically on μ, also D depends analytically on μ. Hence, applying Rouché’s theorem both to the large squares Sk and to the small squares which are sufficiently far away from the origin, it follows that the eigenvalues of the boundary value problem for general g have the same asymptotic distribution as for g=0. Hence Proposition 4 leads to the following.Proposition 6.
Forg∈C1[0,a], there exists a positive integer k0 such that the eigenvalues λk, counted with multiplicity, of the problem (1), (9)–(12), where k∈ℤ∖{0} can be enumerated in such a way that the eigenvalues λk are pure imaginary for |k|<k0, and λ-k=-λk¯ for k≥k0. For k>0, we can write λk=μk2, where the μk have the following asymptotic representation as k→∞:
(42)μk=(k-2)πa+o(1).
In particular, the number of pure imaginary eigenvalues is even.In the remainder of the section, we establish more precise asymptotic expansions of the eigenvalues. For this, it is more convenient to replace(yj)j=14 with the asymptotic fundamental system (ην)ν=14 obtained in [10, Theorem 8.2.1], (1), which can be written as
(43)ην(j)(x,μ)=δν,j(x,μ)eiν-1μx;ν=1,…,4;j=0,…,3,
where
(44)δν,j(x,μ)=[djdxj]{∑r=04‍(μiν-1)-rφr(x)eiν-1μx}e-iν-1μx--+o(μ-4+j),
and [dj/dxj] means that we omit those terms of the Leibniz expansion which contain a function φr(k) with k>4-r. Since the coefficient of y(3) in (1) is zero, we have φ0(x)=1, see [10, (8.2.3)].We will now determine the functionsφ1 and φ2. In this regard, observe that n0=0 and l=4 in the notation of [10, (8.1.2) and (8.1.3)], see [10, Theorem 8.1.2]. From [10, (8.2.45)], we know that
(45)φr=φ1,r=ε1⊤VQ[r]ε1,
where εν is the νth unit vector in ℂ4, V=(i(j-1)(k-1))j,k=14, and Q[r] are 4×4 matrices given by [10, (8.2.28), (8.2.33), and (8.2.34)], that is, Q[0]=I4,
(46)Ω4Q[1]-Q[1]Ω4=Q[0]′=0,Ω4Q[2]-Q[2]Ω4=Q[1]′-14gΩ4εε⊤Ω4-2Q[0],0=εν⊤(Q[2]′+14∑j=12‍k3-jΩ4εε⊤Ω4-1-jQ[2-j])εν--------------------(ν=1,2,3,4),
where k2=-g, k1=-g′, Ω4=diag(1,i,-1,-i) and ε⊤=(1,1,1,1). Letting G(x)=∫0x‍g(t)dt, a lengthy but straightforward calculation gives
(47)φ1=14G,φ2=132G2-18g,
and thus
(48)ην(x,μ)=[(1+14i-ν+1G(x)μ-1---+(-1)ν-1(132G(x)2-18g(x))μ-2)---+{o(μ-2)}∞(1+14i-ν+1G(x)μ-1]eiν-1μx
for ν=1,…,4, where {o(·)}∞ means that the estimate is uniform in x.In view of (44), the characteristic determinant of (1), (9)–(12) with respect to the fundamental system (ηk)k=14 is
(49)D~(μ)=det(Bj(μ2)ηk(·,μ))j,k=14=det(γjkexp(εjk))j,k=14,
where
(50)ε1k=ε2k=0,ε3k=ε4k=ik-1μa,γ1k=δk,2(0,μ)-iαμ2δk,1(0,μ),γ2k=δk,3(0,μ)-g(0)δk,1(0,μ)+iαμ2δk,0(0,μ),γ3k=δk,2(a,μ)+iαμ2δk,1(a,μ),γ4k=δk,3(a,μ)-g(a)δk,1(a,μ)-iαμ2δk,0(a,μ).
An expansion of D~ leads to
(51)D~(μ)=∑m=15‍ψm(μ)eωmμa,
where ω1=1+i, ω2=-1+i, ω3=-1-i, ω4=1-i, ω5=0, and each of the functions ψ1,…,ψ5 has asymptotic representations of the form ckμk+ck-1μk-1+⋯+ck0μk0+o(μk0).It follows from (51) that
(52)D1(μ)≔D~(μ)e-ω1μa=ψ1(μ)+∑m=25‍ψm(μ)e(ωm-ω1)μa,
where ω2-ω1=-2, ω3-ω1=-2-2i, ω4-ω1=-2i, ω5-ω1=-1-i. If argμ∈[-(3π/8),π/8], we have |e(ωm-ω1)μa|≤e-sin(π/8)|μ|a for m=2,3,5 and the terms ψm(μ)e(ωm-ω1)μa for m=2,3,5, can be absorbed by ψ1(μ) as they are of the form o(μ-s) for any integer s. Hence, for argμ∈[-(3π/8),π/8],
(53)D1(μ)=ψ1(μ)+ψ4(μ)e(ω4-ω1)μa=ψ1(μ)+ψ4(μ)e-2iμa,
where
(54)ψ1(μ)=[γ13γ24-γ14γ23][γ31γ42-γ32γ41],ψ4(μ)=[γ12γ23-γ13γ22][γ31γ44-γ34γ41].
A straightforward calculation gives that
(55)γ13γ24-γ14γ23=-2αμ6-(1+α2)(1-i)μ5+2iαμ4+o(μ4),γ31γ42-γ32γ41=2αμ6+(1-i)μ5(1+α2+2αφ1(a))-2iμ4(α(1+φ1(a)2)+(1+α2)φ1(a))+o(μ4),γ12γ23-γ13γ22=-2αμ6+(1+α2)(1+i)μ5-2iαμ4+o(μ4),γ31γ44-γ34γ41=-2αμ6+(1+i)μ5(1+α2-2αφ1(a))-2iμ4(α(1+φ1(a)2)-(1+α2)φ1(a))+o(μ4).
Hence,
(56)μ-12ψ1(μ)=-4α2-(1-i)α[4(1+α2)+αG(a)]μ-1=+i[14α2G2(a)+2α(1+α2)G(a)=====+2(1+α2)2+8α214]μ-2+o(μ-2),(57)μ-12ψ4(μ)=4α2-(1+i)α[4(1+α2)-αG(a)]μ-1=+i[14α2G2(a)-2α(1+α2)G(a)====+2(1+α2)2+8α214]μ-2+o(μ-2).We know by Proposition6 that the zeros μk of D~ satisfy the asymptotic representations μk=k(π/a)+τ0+o(1) as k→∞. In order to improve on these asymptotic representations, write
(58)μk=kπa+τ(k),τ(k)=∑m=02‍τmk-m+o(k-2),========================k=1,2,….
Because of the symmetry of the eigenvalues, we will only need to find the asymptotic expansions as k→∞. We know from Proposition 6 that τ0=-(2π/k), and it is our aim to find τ1 and τ2. To this end, we will substitute (58) into D1(μk)=0, and we will then compare the coefficients of k0, k-1, and k-2.Observe that(59)e-2iμka=e-2iτ(k)a=e-2iτ0aexp(-2ia(τ1k+τ2k2+o(k-2)))=e-2iτ0a×(1-2iaτ11k-(2a2τ12+2iaτ2)1k2+o(k-2)),
while
(60)1μk=aπk(1+aτ(k)kπ)-1=akπ-a2τ0k2π2+o(k-2).
Using (53), D1(μk)=0 can be written as
(61)μk-12ψ1(μk)+μk-12ψ4(μk)e-2iμka=0.
Substituting (56), (57), (59), and (60) into (61) and comparing the coefficients of k0, k-1, and k-2, we getTheorem 7.
Forg∈C1[0,a], there exists a positive integer k0 such that the eigenvalues λk, k∈ℤ, counted with multiplicity, of the problem (1), (9)–(12), where k∈ℤ∖{0} can be enumerated in such a way that the eigenvalues λk are pure imaginary for |k|<k0, and λ-k=-λk¯ for k≥k0, where λk=μk2 and the μk have the asymptotic representations
(62)μk=kπa+τ0+τ1k+τ2k2+o(k-2),
and the numbers τ0, τ1, τ2 are as follows:
(63)τ0=-2πa,τ1=(1+α2)iπα+14G(a)π,τ2=2(1+α2)iπα-12a(1-α2)2π2α2+12G(a)π.
In particular, the number of pure imaginary eigenvalues is even.Remark 8.
In [5], we have considered the differential equation (1) with the same boundary conditions B3, B4 at a as in this paper but only one λ-dependent boundary conditions at 0. We observe that the first two terms in the eigenvalue expansion coincide with those in Case 1 of [5], which differs from the present case that the λ-term is absent in the boundary condition (10). However, the third and fourth terms are similar but different.



---

*Source: 280970-2013-06-12.xml*