An Inverse Spectral Problem for Sturm – Liouville Operators with Singular Potentials on Graphs with a Cycle

This paper is devoted to the solution of inverse spectral problems for Sturm – Liouville operators with singular potentials from class W−1 2 on graphs with a cycle. We consider the lengths of the edges of investigated graphs as commensurable quantities. For the spectral characteristics, we take the spectra of specific boundary value problems and special signs, how it is done in the case of classical Sturm – Liouville operators on graphs with a cycle. From the spectra, we recover the characteristic functions using Hadamard’s theorem. Using characteristic functions and specific signs from the spectral characteristics, we construct Weyl functions (m-function) on the edges of the investigated graph. We show that the specification of Weyl functions uniquely determines the coefficients of differential equation on a graph and we obtain a constructive procedure for the solution of an inverse problem from the given spectral characteristics. In order to study this inverse problem, the ideas of spectral mappings method are applied. The obtained results are natural generalizations of the well-known results of on solving inverse problems for classical differential operators.


INTRODUCTION
The paper concerns the theory of inverse spectral problems for differential operators on geometrical graphs. The inverse problem consists in recovering the potential from the given spectral characteristics. Differential operators on graphs are intensively studied by mathematicians in recent years and have applications in different branches of science and engineering. The inverse problem for the classical Sturm -Liouville operator on an interval has been studied comprehensively in the papers [1][2][3][4]. The case of inverse problem for Sturm -Liouville operators with potentials from class −1 2 , which we call the singular potentials, on an interval was extensively studied in [5][6][7]. The inverse problems for a classical Sturm -Liouville operator on graphs was investigated in many papers [8][9][10][11][12][13][14]. The main result for such operators was obtained in [14], where the arbitrary graph has been considered. The case of inverse problem for Sturm -Liouville operators with singular potentials on graphs is more difficult for investigation, and nowadays there is only a number of papers in this area. The inverse problem on startype graph with such type of potentials has been studied in [15]. Also, some specific types of graphes have been considered in papers [16,17]. The inverse spectral problem for Sturm -Liouville operators with singular potentials on a graph with a cycle has not been studied yet because of the procedure of recovering characteristic functions from the spectra. In this paper we consider the solution of an inverse spectral problem for Sturm -Liouville differential operators with singular potentials on compact graphs with a cycle. As the spectral characteristics we consider the eigenvalues of specific boundary value problems and specific signs, as it is done in [9]. The lengths of the edges of a graph we consider as commensurable quantities. We provide a constructive procedure for the solution of inverse problem from given spectrums.
Let be a graph with a set of vertices { } =0 and a set of edges { } =0 , = [ 0 , ], where edge 0 generates a cycle. We suppose that the length of edge is equal to | |. We consider each edge as a segment [0, | |] and parameterize it by the parameter ∈ [0, | |]. It is convenient for us to choose the orientation such that = | | corresponds to the vertex 0 . We consider lengths of edges , = 0, , as commensurable quantities.
Graph is partitioned into two parts by the vertex 0 : = 0 ∪ , where 0 is a startype graph and is a cycle, generated by edge 0 . Let ( ) Let ℓ( ), = 0, , be the boundary value problem for equation (1) on with Dirichlet boundary condition in the end points of edge , and let ℓ ( ) be boundary value problem for equation (1) on with boundary conditions [1] The inverse problem is formulated as follows.
Inverse Problem 1. Given the Λ, Λ , = 1, , Ω, construct the potential on . The paper is structured as follows. Section 1 contains some auxiliary propositions. Section 2 is devoted to the solution of so-called local inverse problems and a solution of the global inverse problem on a graph.
As in the classical case [13] one can show that functions ( ), = 1, are meromorphic in , namely: Denote by the class of Paley -Wiener functions of exponential type not greater than ∈ R, belonging to 2 (R). It follows from [5][6][7] that where , ( ) ∈ | | are even functions and , ( ) ∈ | | are odd functions. Clearly, Using (10), we obtain We consider the solutions of equation ( We consider the representation Substituting (13) into (2) and (5), we obtain the system of linear equations with variables ( ) and ( ). The determinant of this system we define as ∆ ( , ).

Lemma 2. Following representation is valid
Proof. We define variable in the internal vertex 0 . Then ∆ ( , ) is the determinant of this system Using Laplace expansion for rows, which defines the matching condition for edge 0 , we obtain where 0 is a star-type graph. Let us define the determinant of the system of linear equations by ( ) with variables , = 1, , . We add each -th column to ( + )-th column, = 1, , and get ∆ ( , 0 ) = 1 ( ). Using Laplace expansion for ( ), < , for the first row, we obtain It is obvious, that ( ) = (︁ [1] (| |, ) + [1] (| |, ) )︁ . Using mathematical induction, one can show, that for 1 following representations are valid: Substitute this representation into (15), we obtain (14). Taking into account (12) and properties of the function ( ) ∈ , one can show Then following representation is valid: From [18] one can obtain the following lemma.

Lemma 3.
For sufficiently large | |, such that ∈ ( 0 ), 0 is fixed, following estimate is valid The following lemma describes the asymptotic representations of the solutions .

Lemma 4.
For fixed ∈ [0, | |] and for ∈ ( 0 ), where 0 is fixed and → ∞, following estimates are valid: Proof. Using (13) by Cramer's rule, we obtain where ( ) and ( ) are determinants of the matrices, formed by replacing the corresponding column by the column of free terms.

Corollary 2. Following representation is valid
The eigenvalues Λ can be numbered as { } =0, 0 −1, ∈N ∪ { } = 0 , , ∈Z , where ∈ N, 0 is a multiplicity of a zero eigenvalue of the boundary value problem ( ) with zero potential. Analogously, the eigenvalues Λ can be numbered as where ∈ N, 0 is a multiplicity of zero eigenvalue of the boundary value problem ( ) with zero potential. From [19] it follows, that characteristic functions ∆( , ( )) and ∆( , ( )), = 1, , can be constructed from spectra by Hadamard's theorem: Theorem 1. The specification of spectrums Λ and Λ uniquely determines the characteristic functions respectively by the formula

SOLUTION OF THE INVERSE PROBLEM
Fix , ∈ 1, and consider the following auxiliary inverse problem. Local inverse problems ( , ): Given ( ), construct ( ), ∈ [0, | |]. Everywhere below if a symbol denotes an object, related to , theñ︀ will denote the analogous object, related tõ︀ andˆ= −̃︀. Using the properties of functions from class , analogous to [15] one can prove the following theorem: Everywhere below we chose contour ( ) such thatˆ( ) ∈ 2 ( ). Analogous to [15], one can obtain the main equation