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

Abstract

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameter 𝜆 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.

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 Section 2, 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 Pencil 𝐿

On the interval [0,𝑎], we consider the boundary value problem 𝑦⁽⁴⁾−𝑔𝑦^′′=𝜆²𝐵𝑦,_𝑗(𝜆)𝑦=0,𝑗=1,2,3,4, (1)(2) where 𝑔∈𝐶¹[0,𝑎] is a real valued function and (2) is separated boundary conditions with the 𝐵_𝑗 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 𝐵_𝑗(𝜆)𝑦=𝑦^[𝑝𝑗]𝑎_𝑗+𝑖𝛼𝜀_𝑗𝜆𝑦^[𝑞𝑗]𝑎_𝑗, (3) where 𝑎_𝑗=0 for 𝑗=1,2 and 𝑎_𝑗=𝑎 for 𝑗=3,4, 0≤𝑞_𝑗<𝑝_𝑗≤3, 𝛼>0, and 𝜀_𝑗∈ℂ⧵{0}. We recall that the quasi-derivatives associated with (1) are given by 𝑦^[0]=𝑦,𝑦^[1]=𝑦^′,𝑦^[2]=𝑦^′′,𝑦^[3]=𝑦⁽³⁾−𝑔𝑦^′,𝑦^[4]=𝑦⁽⁴⁾−𝑔𝑦^′′, (4) 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 parameter 𝜆 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 𝑞_𝑗<𝑝_𝑗 for 𝑗=1,…,4.

Further assumptions on the 𝑝_𝑗, 𝑞_𝑗, and 𝜀_𝑗 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 by 𝑈 the collection of the boundary conditions (2) and define the following operators related to 𝑈: 𝑈₀𝑦=𝑦^[𝑝𝑗]𝑎_𝑗⁴_𝑗=1,𝑈₁𝑦=𝜀_𝑗𝑦^[𝑞𝑗]𝑎_𝑗⁴_𝑗=1,𝑦∈𝑊²₄(0,𝑎), (5) where 𝑊²₄(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 𝒟(𝐴(𝑈=𝑦𝑈))𝑦=₁𝑦∶𝑦∈𝑊²₄(),𝒟(𝐾)(𝑀)0,𝑎=𝒟=𝐿₂(0,𝑎)⊕ℂ⁴, (6) given by (𝐴(𝑈𝑦))𝑦=^[4]𝑈₀𝑦(𝐴(𝑈,.for𝑦∈𝒟))𝐾=000𝐼,𝑀=𝐼000 (7) 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 𝐿(𝜆,𝛼)=𝜆²𝑀−𝑖𝛼𝜆𝐾−𝐴(𝑈),𝜆∈ℂ, (8) 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 operator 𝐴(𝑈) associated with (1), (2) is self-adjoint if and only if for 𝑗=1,…,4; the numbers 𝑝_𝑗, 𝑞_𝑗, and 𝜀_𝑗 satisfy the following conditions: 𝑝_𝑗+𝑞_𝑗=3, 𝜀_𝑗=1 if 𝑞_𝑗 is even in case 𝑎_𝑗=0 or odd in case 𝑎_𝑗=𝑎, and 𝜀_𝑗=−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, 𝐵₁(𝜆)𝑦=𝑦^′′(0)−𝑖𝛼𝜆𝑦^′(0),𝐵₂(𝜆)𝑦=𝑦^[3](0)(0),𝐵+𝑖𝛼𝜆𝑦₃(𝜆)𝑦=𝑦^′′(𝑎)+𝑖𝛼𝜆𝑦^′(𝑎),𝐵₄(𝜆)𝑦=𝑦^[3](𝑎)−𝑖𝛼𝜆𝑦(𝑎). (9)(10)(11)(12) As in [4, Proposition 2.3], we obtain the following.

Proposition 2. The operator pencil 𝐿(⋅,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 for 𝑔=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 𝑦_𝑗(⋅,𝜆), 𝑗=1,…,4, of (1) with 𝑦_𝑗^(𝑚)(0)=𝛿_𝑗,𝑚+1 for 𝑚=0,…,3. It is well known that the functions 𝑦_𝑗(⋅,𝜆) are analytic on ℂ with respect to 𝜆. Putting 𝑀(𝜆)=𝐵_𝑖(𝜆)𝑦_𝑗(⋅,𝜆)⁴_𝑖,𝑗=1, (13) 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].

Setting 𝜆=𝜇² and 𝑦=1𝑥,𝜇2𝜇³−1sinh𝜇𝑥2𝜇³sin𝜇𝑥, (14) it is easy to see that 𝑦_𝑗(𝑥,𝜆)=𝑦^(4−𝑗)𝑥,𝜇,𝑗=1,…,4. (15)

The first and the second rows of 𝑀(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, 𝑀(𝜆)=0−𝑖𝛼𝜇²10𝑖𝛼𝜇²𝐵001₃𝜇²𝑦₁𝐵₃𝜇²𝑦₂𝐵₃𝜇²𝑦₃𝐵₃𝜇²𝑦₄𝐵₄𝜇²𝑦₁𝐵₄𝜇²𝑦₂𝐵₄𝜇²𝑦₃𝐵₄𝜇²𝑦₄. (16) An expansion of det𝑀(𝜆)=𝜙(𝜇) gives 𝜙𝜇=−𝛼²𝜇⁴Φ₁𝜇+𝑖𝛼𝜇²Φ₂𝜇+𝑖𝛼𝜇²Φ₃𝜇+Φ₄𝜇, (17) where Φ_𝑗𝜇𝐵=det³𝜇²𝑦_𝜎𝑗,1⋅,𝜇𝐵₃𝜇²𝑦_𝜎𝑗,2𝐵⋅,𝜇₄𝜇²𝑦_𝜎𝑗,1⋅,𝜇𝐵₄𝜇²𝑦_𝜎𝑗,2⋅,𝜇, (18) with 𝜎_𝑗,1,𝜎_𝑗,2=()()()()3,4if𝑗=1,1,3if𝑗=2,2,4if𝑗=3,1,2if𝑗=4. (19) In view of (11), (12), we get that Φ_𝑗𝜇=𝑖𝛼𝜇²𝑦^′_𝜎𝑗,1𝑎,𝜇𝑦_𝜎⁽³⁾_𝑗,2𝑎,𝜇−𝑦^′_𝜎𝑗,2𝑎,𝜇𝑦_𝜎⁽³⁾_𝑗,1𝑎,𝜇+𝑦_𝜎^′′_𝑗,2𝑎,𝜇𝑦_𝜎𝑗,1𝑎,𝜇−𝑦_𝜎^′′_𝑗,1𝑎,𝜇𝑦_𝜎𝑗,2𝑎,𝜇+𝛼²𝜇⁴𝑦^′_𝜎𝑗,1𝑎,𝜇𝑦_𝜎𝑗,2𝑎,𝜇−𝑦^′_𝜎𝑗,2𝑎,𝜇𝑦_𝜎𝑗,1𝑎,𝜇+𝑦_𝜎^′′_𝑗,1𝑎,𝜇𝑦_𝜎⁽³⁾_𝑗,2𝑎,𝜇−𝑦_𝜎^′′_𝑗,2𝑎,𝜇𝑦_𝜎⁽³⁾_𝑗,1𝑎,𝜇. (20) Observing that the 𝑦_𝑗 are given by (14) and (15) a straightforward calculation leads to Φ₁𝜇=12+1𝑖𝛼𝜇sin𝜇𝑎cosh𝜇𝑎+cos𝜇𝑎sinh𝜇𝑎−12𝜇𝑖𝛼sin𝜇𝑎cosh𝜇𝑎−cos𝜇𝑎sinh𝜇𝑎2𝛼²+11−cos𝜇𝑎cosh𝜇𝑎2,Φ1+cos𝜇𝑎cosh𝜇𝑎₂𝜇=−𝑖𝛼𝜇⁴sin𝜇𝑎sinh𝜇𝑎+𝑖𝛼𝜇²−1cos𝜇𝑎cosh𝜇𝑎21+𝛼²×𝜇³,Φsin𝜇𝑎cosh𝜇𝑎+cos𝜇𝑎sinh𝜇𝑎₃𝜇=𝑖𝛼𝜇²−1cos𝜇𝑎cosh𝜇𝑎+𝑖𝛼sin𝜇𝑎sinh𝜇𝑎21+𝛼²,Φ×𝜇sin𝜇𝑎cosh𝜇𝑎−cos𝜇𝑎sinh𝜇𝑎₄𝜇1=−2𝑖𝛼𝜇⁵−1sin𝜇𝑎cosh𝜇𝑎+cos𝜇𝑎sinh𝜇𝑎2𝑖𝛼𝜇³−1sin𝜇𝑎cosh𝜇𝑎−cos𝜇𝑎sinh𝜇𝑎2𝛼²𝜇⁴+11+cos𝜇𝑎cosh𝜇𝑎2𝜇⁴1−cos𝜇𝑎cosh𝜇𝑎. (21) It follows that the term with the highest 𝜇-power in 𝜙 comes from Φ₂ and is a nonzero multiple of 𝜙₀𝜇∶=𝜇⁶sin𝜇𝑎sinh𝜇𝑎. (22)

The following result on the zeros of 𝜙₀, with proper counting, is obvious.

Lemma 3. 𝜙₀ has a zero of multiplicity 8 at 0, simple zeros at 𝜇_𝑘=()𝜋𝑘−2𝑎,𝑘=3,4,…, (23) simple zeros at −𝜇_𝑘, 𝜇_−𝑘=𝑖𝜇_𝑘 and −𝑖𝜇_𝑘 for 𝑘=3,4…, and no other zeros.

Proposition 4. For 𝑔=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=𝜇²_𝑘, where the 𝜇_𝑘 have the following asymptotic representation as 𝑘→∞: 𝜇_𝑘=()𝜋𝑘−2𝑎+𝑜(1). (24) In particular, the number of pure imaginary eigenvalues is even.

Proof. It follows from (17) and (21) that 𝜙𝜇=𝛼²𝜇⁶sin𝜇𝑎sinh𝜇𝑎−𝑖𝛼³+𝛼×𝜇⁵sin𝜇𝑎cosh𝜇𝑎+cos𝜇𝑎sinh𝜇𝑎−𝛼²𝜇⁴1+3cos𝜇𝑎cosh𝜇𝑎−𝛼²𝜇²+1sin𝜇𝑎sinh𝜇𝑎2𝛼⁴𝜇⁴+11−cos𝜇𝑎cosh𝜇𝑎2𝜇⁴1−cos𝜇𝑎cosh𝜇𝑎−𝑖𝛼³+𝛼×𝜇³sin𝜇𝑎cosh𝜇𝑎−cos𝜇𝑎sinh𝜇𝑎. (25) Up to the constant factor 𝛼², the first term equals 𝜙₀(𝜇). Considering that, therefore, 𝜙₁𝜇𝜙𝜇∶=−𝛼²𝜙₀𝜇𝜙₀𝜇 (26) for 𝜙₀(𝜇)≠0, that is, sin(𝜇𝑎)≠0 and sinh(𝜇𝑎)≠0. It follows that 𝜙₁𝜇𝑖𝛼=−³+𝛼𝜇−𝛼cot𝜇𝑎+coth𝜇𝑎₂𝜇²1−𝛼sin𝜇𝑎sinh𝜇𝑎+3cot𝜇𝑎coth𝜇𝑎₂𝜇⁴+𝛼⁴2𝜇²1+1sin𝜇𝑎sinh𝜇𝑎−cot𝜇𝑎coth𝜇𝑎2𝜇²1−𝑖𝛼sin𝜇𝑎sinh𝜇𝑎−cot𝜇𝑎coth𝜇𝑎₃+𝛼𝜇³cot𝜇𝑎−coth𝜇𝑎. (27)
Fix 𝜀∈(0,𝜋/2𝑎), and for 𝑘=3,4,…, let 𝑅_𝑘,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 𝑅_𝑘,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 𝑅_𝑘,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 𝑅_𝑘,𝜀, where 𝑘>𝑘₁(𝜀), the estimate |𝜙₁(𝜇)|<1 holds. By Lemma 3, 𝜙₀ has exactly one simple zero inside 𝑅_𝑘,𝜀. Hence, it follows from Rouché’s theorem that there is exactly one (simple) zero 𝜇_𝑘 of 𝜙 in each 𝑅_𝑘,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 𝜙₀𝜇=𝛼²𝜇⁶sin𝜇𝑎sinh𝜇𝑎−𝛼²𝜇⁴1+3cos𝜇𝑎cosh𝜇𝑎−𝛼²𝜇²+1sin𝜇𝑎sinh𝜇𝑎2𝛼⁴𝜇⁴+11−cos𝜇𝑎cosh𝜇𝑎2𝜇⁴,𝜙1−cos𝜇𝑎cosh𝜇𝑎₁𝜇=−𝑖𝛼³+𝛼×𝜇⁵𝛼sin𝜇𝑎cosh𝜇𝑎+cos𝜇𝑎sinh𝜇𝑎−𝑖³+𝛼×𝜇³sin𝜇𝑎cosh𝜇𝑎−cos𝜇𝑎sinh𝜇𝑎. (28) Then, 𝜙(𝜇)=𝜙₀(𝜇)+𝜙₁(𝜇), and for all 𝜇∈ℂ, 𝜙𝑖𝜇=𝜙₀𝜇−𝜙₁𝜇=𝜙𝜇. (29) 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 of 𝜙. To this end, we are going to estimate 𝜙₁ on the squares 𝑆_𝑘, 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 =𝑒coth𝜇𝑎^{(𝑎𝑥+𝑖𝑎𝑦)}+𝑒^{−(𝑎𝑥+𝑖𝑎𝑦)}𝑒^{(𝑎𝑥+𝑖𝑎𝑦)}−𝑒^{−(𝑎𝑥+𝑖𝑎𝑦)}⟶±1 (30) uniformly in 𝑦 as 𝑥→±∞. Hence, there are 𝜉≥1 and 𝑘₀>0 such that for all 𝑘∈ℕ, 𝑘≥𝑘₀ and 𝛾∈ℝ, ((𝜋coth𝑘+1/2))𝑎+𝑖𝛾𝑎≤𝜉. (31) Note that for 𝑘∈ℕ and 𝛾∈ℝ, ((𝜋cot𝑘+1/2))𝑎+𝑖𝛾𝑎=−tan𝑖𝛾𝑎=−𝑖tanh𝛾𝑎. (32) Hence, we have shown that for 𝑘∈ℕ and 𝛾∈ℝ, ((𝜋cot𝑘+1/2))𝑎+𝑖𝛾𝑎≤1. (33) Furthermore, we will make use of the estimates ((𝜋sinh𝑘+1/2))𝑎𝑎≥1+𝑖𝛾sinh𝑘+2𝜋,((𝜋sin𝑘+1/2))𝑎+𝑖𝛾𝑎=cosh𝛾𝑎≥1, (34) 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 𝑘₁≥𝑘₀ such that |𝜙₁(𝜇)|<1 for all 𝜇∈𝑆_𝑘 with 𝑘>𝑘₁. 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 𝑆_𝑘, for 𝑘∈ℕ with 𝑘≥𝑘₁. Since 𝜙₀ has 4𝑘+8 zeros inside 𝑆_𝑘 and thus 4𝑘+8+4 zeros inside 𝑆_𝑘+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=𝜇²_𝑘 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 𝑖^𝑘−1, 𝑘=1,…,4. We can choose 𝐶𝑥,𝜇=diag1,𝜇,𝜇²,𝜇³𝑖^{(𝑘−1)(𝑗−1)4}_𝑘,𝑗=1 (35) according to [10, Theorem 7.2.4.A]. The boundary condition (9)–(12) can be written in the form 𝐵_𝑗(𝜆)𝑦=𝐵_𝑗𝜇𝑦𝑎_𝑗,𝑦^′𝑎_𝑗,𝑦^′′𝑎_𝑗,𝑦⁽³⁾𝑎_𝑗,𝑗=1,2,3,4. (36) Thus, the boundary matrices defined in [10, (7.3.1)] are given by 𝑊⁽⁰⁾𝜇=𝐵₁𝜇𝐵₂𝜇00𝐶,𝑊0,𝜇₍₁₎𝜇=00𝐵₃𝜇𝐵₄𝜇𝐶𝑎,𝜇, (37) where 𝐵₁𝜇=0,−𝑖𝛼𝜇²,𝐵,1,0₂𝜇=𝑖𝛼𝜇²(0),𝐵,−𝑔,0,1₃𝜇=0,𝑖𝛼𝜇²,𝐵,1,0₄𝜇=−𝑖𝛼𝜇²,−𝑔(𝑎),0,1. (38) Choosing 𝐶₂(𝜇)=diag(𝜇³,𝜇³,𝜇³,𝜇³), it follows that 𝐶₂(𝜇)⁻¹𝑊^(𝑗)(𝜇)=𝑊₀^(𝑗)+𝑂(𝜇⁻¹), where 𝑊₀⁽⁰⁾=,𝑊−𝑖𝛼𝛼𝑖𝛼−𝛼1−𝑖−1𝑖00000000₀₍₁₎=.00000000𝑖𝛼−𝛼−𝑖𝛼𝛼1−𝑖−1𝑖 (39) The Birkhoff matrices are 𝑊₀⁽⁰⁾Δ_𝑗+𝑊₀⁽¹⁾𝐼−Δ_𝑗, (40) where Δ_𝑗, 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 (−1)^𝑗−1−𝑖^𝑗𝛼−𝑖^𝑗+1𝛼(−𝑖)^𝑗−1(−𝑖)^𝑗𝑖^𝑗+2𝛼𝑖^𝑗+3𝛼(−𝑖)^𝑗+1(−𝑖)^𝑗+2=(−1)^𝑗−1()=()−2𝛼)(2𝛼−1^𝑗4𝛼²≠0. (41) Thus, the problem (1), (9)–(12) is Birkhoff regular for 𝛼>0.

5. Asymptotic Expansions of Eigenvalues

With 𝜆=𝜇², 𝐷(𝜇)=det(𝐵_𝑖(𝜇²)𝑦_𝑗(⋅,𝜇))⁴_𝑖,𝑗=1 defines a characteristic determinant of the problem (1), (9)–(12) with respect to the fundamental system 𝑦_𝑗, 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 𝑅_𝑘,𝜀, −𝑅_𝑘,𝜀, 𝑖𝑅_𝑘,𝜀, −𝑖𝑅_𝑘,𝜀 around the zeros of 𝜙₀, |𝐷(𝜇)−𝜙₀(𝜇)|<|𝜙₀(𝜇)| if |𝜇| is sufficiently large. Since the fundamental system 𝑦_𝑗, 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 𝑆_𝑘 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. For 𝑔∈𝐶¹[0,𝑎], there exists a positive integer 𝑘₀ such that the eigenvalues 𝜆_𝑘, counted with multiplicity, of the problem (1), (9)–(12), where 𝑘∈ℤ⧵{0} 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 𝜆_𝑘=𝜇²_𝑘, where the 𝜇_𝑘 have the following asymptotic representation as 𝑘→∞: 𝜇_𝑘=()𝜋𝑘−2𝑎+𝑜(1). (42) 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 (𝑦_𝑗)⁴_𝑗=1 with the asymptotic fundamental system (𝜂_𝜈)⁴_𝜈=1 obtained in [10, Theorem 8.2.1], (1), which can be written as 𝜂_𝜈^(𝑗)𝑥,𝜇=𝛿_𝜈,𝑗𝑥,𝜇𝑒^{𝑖𝜈−1𝜇𝑥};𝜈=1,…,4;𝑗=0,…,3, (43) where 𝛿_𝜈,𝑗=𝑑𝑥,𝜇^𝑗𝑑𝑥^𝑗4_𝑟=0𝜇𝑖^{𝜈−1−𝑟}𝜑_𝑟(𝑥)𝑒^{𝑖𝜈−1𝜇𝑥}𝑒^{−𝑖𝜈−1𝜇𝑥}+𝑜𝜇^−4+𝑗, (44) and [𝑑^𝑗/𝑑𝑥^𝑗] means that we omit those terms of the Leibniz expansion which contain a function 𝜑_𝑟^(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 functions 𝜑₁ 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 𝜑_𝑟=𝜑_1,𝑟=𝜀^⊤₁𝑉𝑄^[𝑟]𝜀₁, (45) where 𝜀_𝜈 is the 𝜈th unit vector in ℂ⁴, 𝑉=(𝑖^{(𝑗−1)(𝑘−1)})⁴_𝑗,𝑘=1, 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, Ω₄𝑄^[1]−𝑄^[1]Ω₄=𝑄^′[0]Ω=0,₄𝑄^[2]−𝑄^[2]Ω₄=𝑄^′[1]−14𝑔Ω₄𝜀𝜀^⊤Ω₄⁻²𝑄^[0],0=𝜀^⊤_𝜈𝑄^′[2]+14²_𝑗=1𝑘_3−𝑗Ω₄𝜀𝜀^⊤Ω₄^−1−𝑗𝑄^[2−𝑗]𝜀_𝜈(𝜈=1,2,3,4), (46) where 𝑘₂=−𝑔, 𝑘₁=−𝑔^′, Ω₄=diag(1,𝑖,−1,−𝑖) and 𝜀^⊤=(1,1,1,1). Letting 𝐺(𝑥)=∫^𝑥₀𝑔(𝑡)𝑑𝑡, a lengthy but straightforward calculation gives 𝜑₁=14𝐺,𝜑₂=1𝐺32²−18𝑔, (47) and thus 𝜂_𝜈=1𝑥,𝜇1+4𝑖^−𝜈+1𝐺(𝑥)𝜇⁻¹+(−1)^𝜈−11𝐺(𝑥)32²−18𝑔(𝑥)𝜇⁻²+𝑜𝜇⁻²_∞𝑒^{𝑖𝜈−1𝜇𝑥} (48) 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 (𝜂_𝑘)⁴_𝑘=1 is 𝐷𝜇=det𝐵_𝑗𝜇²𝜂_𝑘⋅,𝜇⁴_𝑗,𝑘=1=det𝛾_𝑗𝑘exp𝜀_𝑗𝑘⁴_𝑗,𝑘=1, (49) where 𝜀_1𝑘=𝜀_2𝑘=0,𝜀_3𝑘=𝜀_4𝑘=𝑖^𝑘−1𝛾𝜇𝑎,_1𝑘=𝛿_𝑘,20,𝜇−𝑖𝛼𝜇²𝛿_𝑘,1,𝛾0,𝜇_2𝑘=𝛿_𝑘,30,𝜇−𝑔(0)𝛿_𝑘,10,𝜇+𝑖𝛼𝜇²𝛿_𝑘,0,𝛾0,𝜇_3𝑘=𝛿_𝑘,2𝑎,𝜇+𝑖𝛼𝜇²𝛿_𝑘,1,𝛾𝑎,𝜇_4𝑘=𝛿_𝑘,3𝑎,𝜇−𝑔(𝑎)𝛿_𝑘,1𝑎,𝜇−𝑖𝛼𝜇²𝛿_𝑘,0𝑎,𝜇. (50) An expansion of 𝐷 leads to 𝐷𝜇=⁵_𝑚=1𝜓_𝑚𝜇𝑒^{𝜔𝑚𝜇𝑎}, (51) 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 𝑐_𝑘𝜇^𝑘+𝑐_𝑘−1𝜇^𝑘−1+⋯+𝑐_𝑘0𝜇^𝑘0+𝑜(𝜇^𝑘0).

It follows from (51) that 𝐷₁𝜇∶=𝐷𝜇𝑒^{−𝜔1𝜇𝑎}=𝜓₁𝜇+⁵_𝑚=2𝜓_𝑚𝜇𝑒^{(𝜔𝑚−𝜔1)𝜇𝑎}, (52) 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 𝜓_𝑚(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], 𝐷₁𝜇=𝜓₁𝜇+𝜓₄𝜇𝑒^{(𝜔4−𝜔1)𝜇𝑎}=𝜓₁𝜇+𝜓₄𝜇𝑒^{−2𝑖𝜇𝑎}, (53) where 𝜓₁𝜇=𝛾₁₃𝛾₂₄−𝛾₁₄𝛾₂₃𝛾₃₁𝛾₄₂−𝛾₃₂𝛾₄₁,𝜓₄𝜇=𝛾₁₂𝛾₂₃−𝛾₁₃𝛾₂₂𝛾₃₁𝛾₄₄−𝛾₃₄𝛾₄₁. (54) A straightforward calculation gives that 𝛾₁₃𝛾₂₄−𝛾₁₄𝛾₂₃=−2𝛼𝜇⁶−1+𝛼²(1−𝑖)𝜇⁵+2𝑖𝛼𝜇⁴+𝑜𝜇⁴,𝛾₃₁𝛾₄₂−𝛾₃₂𝛾₄₁=2𝛼𝜇⁶+(1−𝑖)𝜇⁵1+𝛼²+2𝛼𝜑₁(𝑎)−2𝑖𝜇⁴𝛼1+𝜑₁(𝑎)²+1+𝛼²𝜑₁(𝑎)+𝑜𝜇⁴,𝛾₁₂𝛾₂₃−𝛾₁₃𝛾₂₂=−2𝛼𝜇⁶+1+𝛼²(1+𝑖)𝜇⁵−2𝑖𝛼𝜇⁴+𝑜𝜇⁴,𝛾₃₁𝛾₄₄−𝛾₃₄𝛾₄₁=−2𝛼𝜇⁶+(1+𝑖)𝜇⁵1+𝛼²−2𝛼𝜑₁(𝑎)−2𝑖𝜇⁴𝛼1+𝜑₁(𝑎)²−1+𝛼²𝜑₁(𝑎)+𝑜𝜇⁴. (55) Hence, 𝜇⁻¹²𝜓₁𝜇=−4𝛼²−(1−𝑖)𝛼41+𝛼²+𝛼𝐺(𝑎)𝜇⁻¹1+𝑖4𝛼²𝐺²(𝑎)+2𝛼1+𝛼²𝐺(𝑎)+21+𝛼²²+8𝛼²𝜇⁻²+𝑜𝜇⁻²,𝜇⁻¹²𝜓₄𝜇=4𝛼²−(1+𝑖)𝛼41+𝛼²−𝛼𝐺(𝑎)𝜇⁻¹1+𝑖4𝛼²𝐺²(𝑎)−2𝛼1+𝛼²𝐺(𝑎)+21+𝛼²²+8𝛼²𝜇⁻²+𝑜𝜇⁻². (56)(57)

We know by Proposition 6 that the zeros 𝜇_𝑘 of 𝐷 satisfy the asymptotic representations 𝜇_𝑘=𝑘(𝜋/𝑎)+𝜏₀+𝑜(1) as 𝑘→∞. In order to improve on these asymptotic representations, write 𝜇_𝑘𝜋=𝑘𝑎+𝜏(𝑘),𝜏(𝑘)=²_𝑚=0𝜏_𝑚𝑘^−𝑚+𝑜𝑘⁻²,𝑘=1,2,…. (58) 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(𝜇_𝑘)=0, and we will then compare the coefficients of 𝑘⁰, 𝑘⁻¹, and 𝑘⁻².

Observe that 𝑒^{−2𝑖𝜇𝑘𝑎}=𝑒^{−2𝑖𝜏(𝑘)𝑎}=𝑒^{−2𝑖𝜏0𝑎}𝜏exp−2𝑖𝑎¹𝑘+𝜏₂𝑘²+𝑜𝑘⁻²=𝑒^{−2𝑖𝜏0𝑎}×1−2𝑖𝑎𝜏₁1𝑘−2𝑎²𝜏²₁+2𝑖𝑎𝜏₂1𝑘²+𝑜𝑘⁻², (59) while 1𝜇_𝑘=𝑎(𝑘)𝜋𝑘1+𝑎𝜏𝑘𝜋⁻¹=𝑎−𝑎𝑘𝜋²𝜏₀𝑘²𝜋²+𝑜𝑘⁻². (60) Using (53), 𝐷₁(𝜇_𝑘)=0 can be written as 𝜇_𝑘⁻¹²𝜓₁𝜇_𝑘+𝜇_𝑘⁻¹²𝜓₄𝜇_𝑘𝑒^{−2𝑖𝜇𝑘𝑎}=0. (61) Substituting (56), (57), (59), and (60) into (61) and comparing the coefficients of 𝑘⁰, 𝑘⁻¹, and 𝑘⁻², we get

Theorem 7. For 𝑔∈𝐶¹[0,𝑎], there exists a positive integer 𝑘₀ such that the eigenvalues 𝜆_𝑘, 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 𝜆_𝑘 are pure imaginary for |𝑘|<𝑘₀, and 𝜆_−𝑘=−𝜆_𝑘 for 𝑘≥𝑘₀, where 𝜆_𝑘=𝜇²_𝑘 and the 𝜇_𝑘 have the asymptotic representations 𝜇_𝑘𝜋=𝑘𝑎+𝜏₀+𝜏₁𝑘+𝜏₂𝑘²+𝑜𝑘⁻², (62) and the numbers 𝜏₀, 𝜏₁, 𝜏₂ are as follows: 𝜏₀=−2𝜋𝑎,𝜏₁=1+𝛼²𝑖+1𝜋𝛼4𝐺(𝑎)𝜋,𝜏₂=21+𝛼²𝑖−1𝜋𝛼2𝑎1−𝛼²²𝜋²𝛼²+12𝐺(𝑎)𝜋. (63)
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.

Acknowledgments

Various of the above calculations have been verified with Sage. This work is based upon research supported by the National Research Foundation of South Africa under Grant number 80956. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF do not accept any liability in regard thereto.