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Matrix Sturm-Liouville equation with a Bessel-type singularity on a finite interval

The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.


Introduction
Consider the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval Here Y = [y k (x)] m k=1 is a vector function, λ is the spectral parameter, Q(x) and ω are m × m matrices.
In this paper, we construct special fundamental systems of solutions (FSS) for equation (1) and study asymptotic behavior of the spectral data. Our results will be useful in the theory of direct and inverse spectral problems for systems with Bessel-type singularities.
Differential equations with regular singularities are studied intensively in recent years. The direct and inverse spectral problems for scalar analogue of equation (1) (m = 1) have been investigated in [1] and [2], respectively, even for the more general case of the singularities in the both ends of the interval. These problems arose from the analysis of the differential equations with turning points. Such equations have applications in electronics and geophysics (see, for example, [3,4,5]). The case of singularities inside the interval has also been investigated [6,7]. In papers [8,9] the Dirac system with regular singularity was studied. The papers [10,11] are devoted to inverse problems for differential operators with singularities on geometrical graphs.
The approach in the mentioned papers is based on the special FSS, constructed in [12,13] for higher-order differential equations with the regular singularity. The first FSS is formed by Bessel-type solutions, which are analytic in λ and have the prescribed asymptotic behavior at the singular point. The second FSS consists of Birkhoff-type solutions with the known behavior for large values of the spectral parameter. The two FSS are connected by the Stokes multipliers. The asymptotics for the Stokes multipliers play a crucial role in the spectral theory for operators with singularities, since they give an opportunity to analyze the behavior of solutions and spectral characteristics for these operators.
The matrix equation (1) with the Bessel-type singularity have been studied in [15], where inverse scattering problem, related to quantum mechanics, was solved. However, the method of [15], based on special non-integral transforms, removing singularities, works only in some particular cases. In paper [16], the inverse spectral problem for equation (1) on the finite interval was studied in the more general case. The uniqueness theorem was proved.
In the present paper, we continue the research, started in [16]. We provide the construction of the FSS for equation (1) and derive asymptotic formulas for the Stokes multipliers, which appeared in the short note [16] without proofs. Further we set the boundary value problem for equation (1) and obtain the asymptotics for its spectral data, consisting of the eigenvalues and so-called weight matrices. The weight matrices equal to the residues of the Weyl matrix, they generalize the notion of the weight numbers for the scalar Sturm-Liouville equation (see [14]). The spectral data, considered in this paper, are natural spectral characteristics, by which a matrix Sturm-Liouville operator can be recovered. The inverse problem by the spectral data for the matrix Sturm-Liouville equation without singularities was studied, for example, in [17] (see also references therein). The asymptotics for the eigenvalues and for the weight matrices can further be used for the constructive solution of the inverse problem for equation (1). We note that the asymptotic analysis of the spectral data of the considered problem involves difficulties, related with the asymptotic closeness of large eigenvalues. Therefore one can not calculate asymptotics of the weight matrices separately, but can only analyze the sums of the residues, corresponding to some contours. Such approach allowed to solve inverse problems for matrix Sturm-Liouville operators in [17,18].
The paper is organized as follows. For the convenience of the reader, we provide the standard results on the FSS for the scalar Sturm-Liouville equation with the Bessel singularity in Section 2. In Section 3, we construct the Bessel-type and the Birkhoff-type solutions for the matrix equation (1). In Section 4, we establish the connection between two types of the solutions and obtain asymptotics for the Stokes multipliers. In Section 5, the boundary conditions are set for equation (1), and the notion of the spectral data is introduced. We derive asymptotic formulas for the eigenvalues and the weight matrices in Sections 6 and 7, respectively. Let us introduce the notation. Denote ρ := √ λ, Re ρ ≥ 0, τ := Im ρ. Denote the unit m × m matrix by I, its q-th column by e q and its q-th row by e * q . We use the following matrix norm: A = max

Scalar case
In this section, we briefly recall the construction of the FSS for the scalar equation where ω = ν 2 − 1 4 , ν > 0, ν / ∈ N. We adapt the results from [12,13], where they were obtained for the higher-order differential operators.
Let λ = 1. Then equation (2) has a fundamental system of Bessel solutions where The constants c j0 can be chosen arbitrary. Fix them in such a way, that c 10 c 20 = (2ν) −1 . Then Denote by Π the complex x-plane with the cut x ≤ 0. The solutions c j (x) are analytic in Π. Equation (2) with λ = 1 also has Jost solutions Here the integral is taken along the ray {t : arg t = arg x, |t| ∈ (|x|, +∞)}. The functions e 1 (x) and e 2 (x) are analytic in the upper and the lower half-plane, respectively. They can be represented in the form with some constants β 0 kj . Relation (6) gives an analytic continuation of e k (x) into the whole Π. One can show that the following asymptotic relations hold for ν = 0, 1: It follows from (5) and (6), that By virtue of (4), (7) and (6), det[β 0 kj ] k,j=1,2 = −2i. Consequently, form FSS for it. The solutions c j (x, λ) are entire in λ, while e k (x, ρ) are analytic in ρ for Re ρ > 0, continuous for Re ρ ≥ 0, |ρ| ≥ ρ * and satisfy the estimates with some constant M 0 . The relations (6) yield Consequently, in view of (3) and (9), we have 3. Bessel-type and Birkhoff-type solutions for the matrix equation The matrix equation (1) with Q(x) ≡ 0 splits into m scalar equations, and therefore it has matrix solutions where c jq (x, λ) and e jq (x, ρ) are constructed like c j (x, λ) and e j (x, ρ) in Section 2, respectively, with ν q instead of ν. In particular, µ 1q = 1 2 − ν q , µ 2q = 1 2 + ν q , q = 1, m. Let S j (x, λ) and S * j (x, λ), j = 1, 2, be the matrix solutions of the following integral equations where Along with equation (1), consider the following equation where Z = Z(x) is a row vector. The matrix-functions S j (x, λ) and S * j (x, λ), j = 1, 2, are called Bessel-type solutions for equations (1) and (15), respectively. They have the following properties [16]: 1. The columns of the matrices S j (x, λ), j = 1, 2, and the rows of the matrices S * j (x, λ), j = 1, 2, form FSS for equations (1) and (15).
In the next theorem, we construct the so-called Birkhoff-type solutions with the prescribed behavior as |ρ| → ∞.
Note that by virtue of (10) and (12), (20) and (21) can be rewritten in the form The kernel K 1 (x, t, ρ) takes the form x < t < T. Similar representations can be written for K 2 (x, t, ρ). Clearly, Therefore the solutions of (23) can be found by the method of successive approximations for |ρ| > ρ * , if ρ * is sufficiently large. The solutions u k (x, ρ) are analytic in Ω 0 , continuous in Ω 0 and satisfy the estimates Consequently, the conditions (i 1 ) and (i 2 ) are fulfilled for the solutions Y k (x, ρ), k = 1, 2, of (20), (21).

Stokes multipliers
In this section, we investigate the connection between the Bessel-type and the Birkhoff-type solutions. Clearly, there exist matrix coefficients B kj (ρ), called the Stokes multipliers, such that For the case Q(x) ≡ 0, we have Recall that the matrices C j (x, λ) and E k (x, ρ), j, k = 1, 2, are diagonal. In view of (11), The following theorem presents asymptotic formulas for the Stokes multipliers B kj (ρ). These asymptotics play a crucial role in the analysis of direct and inverse problems for equation (1). Further we use these relations in order to derive asymptotic formulas for the solutions S j (x, λ) as |λ| → ∞, and for the spectral characteristics of equation (1).

Theorem 2. The following relations hold
In order to prove Theorem 2, we need the following auxiliary lemma.

Spectral data
In this section, we introduce boundary conditions for equation (1) and the notion of spectral data for the boundary value problem.
Introduce the linear forms In view of (18), we have σ j (S k ) = δ jk I. Note that for the classical matrix Sturm-Liouville equation we have Consider the boundary value problem L = L(Q, h, H) for equation (1) (Φ(x, λ)). is called the Weyl matrix of the problem L. The Weyl matrix is a natural spectral characteristic for matrix Sturm-Liouville operators (see [17,18]). It generalizes the notion of the Weyl function for the scalar case m = 1 (see [19,14]).
Let ϕ(x, λ) be the matrix solution of equation (1) under initial conditions σ 1 (ϕ) = I, The eigenvalues of the boundary value problem L coincide with the zeros of the characteristic function ∆(λ) := V (ϕ(x, λ)). We denote the eigenvalues, counting with their multiplicities, by {λ p } p≥1 , |λ p | ≤ |λ p+1 |. The Weyl matrix M(λ) is meromorphic in λ and its poles coincide with the eigenvalues of L. Assume that all the poles of M(λ) are simple. Note that in this case the multiplicities of the eigenvalues (the number of the corresponding vector eigenfunctions) equal to the multiplicities of the zeros of the analytic function ∆(λ) (the proof is similar to [17,Lemma 4]). We will call the residues α p := Res λ=λp M(λ) the weight matrices. The collection {λ p , α p } p≥1 is called the spectral data of the problem L. One can easily show that Here m p is the multiplicity of the eigenvalue λ p . In [16] the inverse problem is studied, which consists in recovering of the boundary value problem L from the Weyl matrix. In view of (39), this problem is equivalent to the following one.
Inverse problem. Given the spectral data {λ p , α p } p≥1 , determine Q, h and H.
Moreover, we will show that the matrix ω can also be determined uniquely from the spectral data.
We plan to devote a separate paper to the formulated inverse problem. In the next two sections of this paper, we derive asymptotic formulas for λ p and α p . They will be useful for the solution of the inverse problem.

Asymptotics of the eigenvalues
In order to obtain asymptotic formulas for the eigenvalues, we need the following auxiliary result.
Proof. First of all, note that the number p 0 − n 0 m does not depend on the choice of n 0 .

Asymptotics of the weight matrices
Consider the eigenvalues λ nq , n ≥ n 0 , q = 1, m, numbered in accordance with Theorem 3. Let m nq be their multiplicities, and α nq = Res λ=λnq M(λ). Corollary 3. The matrix ω is uniquely determined by the weight matrices {α p } p≥1 or by the Weyl matrix M(λ).