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We study an inverse spectral problem for the Sturm-Liouville operator on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. As spectral characteristics, we consider the spectrum of the main problem together with the spectra of two Dirichlet-Dirichlet problems and one Robin-Dirichlet problem on the edges of the graph and investigate their properties and asymptotic behavior. We prove that if these four spectra do not intersect, then the inverse problem of recovering the operator is uniquely solvable. We give an algorithm for the solution of the inverse problem with respect to this quadruple of spectra.

This paper is devoted to the study of the inverse spectral problem for Sturm-Liouville operators on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. The considered inverse problem consists of recovering the Sturm-Liouville operator on a graph from the given spectral characteristics. Differential operators on graphs (networks, trees) often appear in mathematics, mechanics, physics, geophysics, physical chemistry, electronics, nanoscale technology and branches of natural sciences and engineering (see [

We consider a three-star graph

The problem of small transverse vibrations of a three-star graph consisting of three inhomogeneous smooth strings joined at the internal vertex with two pendent ends fixed and one pendent end which can move without friction in the directions orthogonal to their respective equilibrium positions can be reduced to this problem by the Liouville transformation. This problem occurs also in quantum mechanics when one considers a quantum particle subject to the Shrödinger equation moving in a quasi-one-dimensional graph domain.

In this paper, we study the inverse problem of recovering the potential

Denote by

Let us consider the operator-theoretical interpretation of our problem. Denote by A the operator acting in the Hilbert space

This paper has the following structure: in Section

In this section, we describe the properties of sequences of eigenvalues of the boundary-value problems

Let us denote by

or

Let

the multiplicities are taken into account.

If a sequence has even number of pure imaginary elements, we exclude the index zero from enumeration to make it proper.

Throughout Section

Let us denote by

The functions

Using the formulas of [

The set

In the same way as [

To compare necessary conditions on a sequence to be the spectrum of the boundary-value problem

Let

If

Under the conditions of Theorem

For investigation of direct and inverse spectral problems, methods of the theory of entire and meromorphic functions are widely used. For this reason, we give several notation and definitions for what follows.

If

Let

The pair

Let

A function

The class

It is easy to check that

An entire function

all the zeros of

for some

the type of

Let us introduce the entire functions

Using (

(1) the zeros of the functions

(2) the functions

The zeros of

The functions

Let

The functions

By virtue of the formulas (

Now using (

The sequences

the maximal multiplicity of

Denote

In the present section, we study the problem of reconstruction of the potential

the operator

Let the following conditions be satisfied.

Three sequences

one has

A sequence

The sequences

Denote by

Let us set

One has

Substituting (

Now since the functions

The following inequalities are valid:

In the same way as proof of [

Let

If condition 1(ii) of Theorem

This research is done with financial support of Research Office of the University of Tabriz.