Asymptotic Formulae for Weight Numbers of the Sturm– Liouville Boundary Problem on a Star-shaped Graph

In this article the Sturm – Liouville boundary value problem on the graph Γ of a special structure is considered. The graph Γ has m edges, joined at one common vertex, and m vertices of degree 1. The boundary value problem is set by the Sturm – Liouville differential expression with real-valued potentials, the Dirichlet boundary conditions, and the standard matching conditions.This problem has a countable set of eigenvalues. We consider the so-called weight numbers, being the residues of the diagonal elements of the Weyl matrix in the eigenvalues. These elements are meromorphic functions with simple poles which can be only the eigenvalues. We note that the considered weight numbers generalize the weight numbers of Sturm – Liouville operators on a finite interval, equal to the reciprocals to the squared norms of eigenfunctions. These numbers together with the eigenvalues play a role of spectral data for unique reconstruction of operators.We obtain asymptotic formulae for the weight numbers using the contour integration, and in the case of the asymptotically close eigenvalues the formulae are got for the sums. The formulae can be used for the analysis of inverse spectral problems on the graphs.


INTRODUCTION
Consider the graph Γ which consists of m edges e j , m 2, j = 1, m, joined at a common vertex. Let the graph Γ be parametrized so that x j ∈ [0, π] where the parameter x j corresponds to the edge e j , the parameter x j = 0 in the boundary vertex and x j = π in the common vertex, j = 1, m. We call Γ a star-shaped graph.
A function on the graph is a vector function where the components y j are functions on the edges e j correspondingly, y j (x j ) ∈ W 2 2 [0, π], j = 1, m. We denote by g ′ differentiation of the function g with respect to the first argument. Consider the differential expression Then the Sturm -Liouville boundary value problem on the graph can be written as follows: y j (0) = 0, j = 1, m, m j=1 y ′ j (π) = 0, y 1 (π) = y 2 (π) = . . . = y m (π), c Кузнецова М. А., 2018 where λ is the spectral parameter, the equalities (3) are Dirichlet conditions, and (4)- (5) are the standard matching conditions. In (1) the functions q j (x j ) are called potentials, q j (x j ) ∈ L 2 [0, π], q j (x j ) ∈ R.
The differential operator L, given by the differential expression (1) and the conditions (3)- (5), is self-adjoint in the corresponding Hilbert space (see [1] for details).
Since the differential operators on the graphs have applications in physics, chemistry, nano-technology, they are studied actively (see [2,3]). In the article we obtain asymptotic formulae for weight numbers of the problem (2)-(5) which generalize the weight numbers on a finite interval [4,Chapter 1]. Those asymptotic formulae can be applied for studying of inverse spectral problems for differential operators on graphs. Weight numbers together with eigenvalues have been used for reconstruction of the potentials of the Sturm -Liouville operators on graphs, e.g., in [5,6].
The difficult case is when the eigenvalues are asymptotically close though not multiple. The asymptotic formulae are got by using the integration over the contours, containing the asymptotically close eigenvalues, in the plain of the spectral parameter.Thus, the asymptotic formulae are obtained for the sums of the weight numbers, as it has been done in [7] for the weight matrices for the matrix Sturm -Liouville operator.

PRELIMINARIES
In this section we introduce a characteristic function of the operator L, the zeros of which coincide with the eigenvalues. We also provide auxiliary results from [8,9], related to the eigenvalues of L.
As S j (π, λ), S ′ j (π, λ) are entire functions of λ, the function ∆(λ) is also entire. Reconstructing the determinant in (13), we obtain Lemma 1. The number λ 0 is an eigenvalue of problem (2)-(5) of multiplicity k if and only if λ 0 is a zero of characteristic function of multiplicity k.
The statement of the Lemma 1 results from the self-adjointness of L and is proved with the same technique as in [7,Lemma 3]. From the self-adjointness of L it also follows that the eigenvalues of the boundary problem (2)-(5) are real. Denote j=1 ω j /m. We will mean by {κ n } ∞ n=1 different sequences from l 2 . Considering these designations, the following theorem can be formulated: The operator L has a countable set of the eigenvalues. The eigenvalues can be enumerated in such way that the further formulae are satisfied: where the square root is taken according the same rule as in (6).

MAIN RESULTS
In that chapter we define and study weight numbers based on the Weyl matrix.
is called the Weyl matrix and generalize the notion of the Weyl function for differential operators on intervals (see [4]). Weyl functions and their generalizations are natural spectral characteristics, often used for reconstruction of operators. A system of 2m columns of the matrix solutions C(λ), S(λ) is fundamental, and one can show, that M (λ) = (V (S(λ))) −1 V (C(λ)).
In view of (17) the elements of the matrix M (λ) = {M kl (λ)} m k,l=1 can be calculated as The elements of the matrix M (λ) are meromorphic functions, and their poles may be only zeros of the characteristic function ∆(λ). Moreover, analogously to [7, Lemma 3], we prove the following lemma. It is sufficient to prove that for any element of A(λ) the number λ 0 can not be a pole of order greater than 1, where The number λ 0 is zero of the numerator of multiplicity not less then b − 1. From that the statement of the theorem follows. We introduce the constants α k jn = Res λ=λ (j) n M kk (λ) which are called weight numbers.
We also mean by {κ n (z)} ∞ n=1 different sequences of continuous functions such that The main results of the paper are stated in the following two theorems.
Formula (20) is proved analogously. where , and the product over empty set is understood as 1.
Proof. Denote by r such positive number that the circle |z − z (s) | r does not contain z (j) , j ∈ N (s) and |z (s) | + r < R, r C > 0. We call the circumference of that circle γ(s). The following analogue of the formulae (28) can be proved: We designate .
As ω j −ω jn = κ n and the coefficients of where z ∈ γ(s). We integrate the fraction F p (z).
Further, let b(s) = 1. When z (s) is a zero of f (z), computations are the same as in the case b(s) > 1. So we assume f (z (s) ) = 0, and consequently p ∈ W (s). Rewriting F p (z) as and integrating over γ(s), we obtain (30).