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

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.


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 selfadjoint, 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 .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  = 0.In Section 4, we prove that the boundary value problem under investigation is Birkhoff regular, which implies that the eigenvalues for general  are small perturbations of the eigenvalues for  = 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].

The Quadratic Operator Pencil 𝐿
On the interval [0, ], we consider the boundary value problem where  ∈  1 [0, ] is a real valued function and ( 2) is separated boundary conditions with the   depending linearly on the eigenvalue parameter .The boundary conditions ( 2) are taken at the endpoint 0 for  = 1, 2 and at the endpoint  > 0 for  = 3, 4. Further, we assume for simplicity that where   = 0 for  = 1, 2 and   =  for  = 3, 4, 0 ≤   <   ≤ 3,  > 0, and   ∈ C \ {0}.We recall that the quasi-derivatives associated with (1) are given by  [0] = ,  [1] =   ,  [2] =   ,  [3] =  (3) −   ,  [4] =  (4) see [9, page 26].In order to have independent boundary conditions, we will also assume that the numbers  1 ,  1 ,  2 ,  2 as well as the numbers  3 ,  3 ,  4 ,  4 are mutually disjoint.Recall that in applications, using separation of variables, the parameter  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   <   for  = 1, . . ., 4.
Further assumptions on the   ,   , and   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.
As in [4, Proposition 2.3], we obtain the following.
Proposition 2. The operator pencil (⋅, ) is a Fredholm valued operator function with index 0.The spectrum of the Fredholm operator (⋅, ) consists of discrete eigenvalues of finite multiplicities, and all eigenvalues of (⋅, ),  ≥ 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  ∈ C, (, ) is a relatively compact perturbation of (0, 0), where (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].

Asymptotics of Eigenvalues for 𝑔 = 0
In this section, we investigate the boundary value problem ( 1), ( 2) with  = 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 .Observe that for  = 0, the quasi-derivatives  [𝑗] coincide with the standard derivatives  (𝑗) .We take the canonical fundamental system   (⋅, ),  = 1, . . ., 4, of (1) with  ()  (0) =  ,+1 for  = 0, . . ., 3. It is well known that the functions   (⋅, ) are analytic on C with respect to .Putting the eigenvalues of the boundary value problem ( 1), ( 2) are the eigenvalues of the analytic matrix function , where the corresponding geometric and algebraic multiplicities coincide, see [10, Theorem 6.3.2].
Setting  =  2 and it is easy to see that The first and the second rows of () have exactly two nonzero entries (for  ̸ = 0), and these nonzero entries are: ) . (16) An expansion of det () = () gives where with In view of (11), (12), we get that Observing that the   are given by ( 14) and (15) a straightforward calculation leads to It follows that the term with the highest -power in  comes from Φ 2 and is a nonzero multiple of  0 () :=  6 sin () sinh () .
The following result on the zeros of  0 , with proper counting, is obvious.Proposition 4. For  = 0, there exists a positive integer  0 such that the eigenvalues λ ,  ∈ Z \ {0}, counted with multiplicity, of the problem (1), ( 9)-( 12) can be enumerated in such a way that the eigenvalues λ are pure imaginary for || <  0 , and λ− = − λ for  ≥  0 .For  > 0, we can write that λ = μ2  , where the μ have the following asymptotic representation as  → ∞: In particular, the number of pure imaginary eigenvalues is even.
Proof.It follows from ( 17) and (21) that Up to the constant factor  2 , the first term equals  0 ().

Birkhoff Regularity
We refer to [10, Definition 7.3.1]for the definition of Birkhoff regularity.

Asymptotic Expansions of Eigenvalues
With  =  2 , () = det(  ( 2 )  (⋅, )) 4 ,=1 defines a characteristic determinant of the problem (1), ( 9)-( 12) with respect to the fundamental system   ,  = 1, 2, 3, 4 considered in Section 3. Observe that  is the corresponding characteristic determinant for  = 0. Due to the Birkhoff regularity,  only influences lower order terms in .Together with the estimates in Section 3, it can be inferred that outside the interior of the small squares  , , − , ,  , , − , around the zeros of  0 , |() −  0 ()| < | 0 ()| if || is sufficiently large.Since the fundamental system   ,  = 1, 2, 3, 4, depends analytically on , also  depends analytically on .Hence, applying Rouché's theorem both to the large squares   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  have the same asymptotic distribution as for  = 0. Hence Proposition 4 leads to the following.Proposition 6.For  ∈  1 [0, ], there exists a positive integer  0 such that the eigenvalues   , counted with multiplicity, of the problem (1), ( 9)-( 12), where  ∈ Z \ {0} can be enumerated in such a way that the eigenvalues   are pure imaginary for || <  0 , and  − = −  for  ≥  0 .For  > 0, we can write   =  2  , where the   have the following asymptotic representation as  → ∞: In particular, the number of pure imaginary eigenvalues is even.