On Recovering Differential Operators on a Closed Set from Spectra

The Sturm – Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points. In order to solve the inverse spectral problem for this class of closed sets, we develop ideas of the method of spectral mappings. We also establish and use connections between the Weyl-type functions related to different subsets of the main closed set. Using these ideas and properties we obtain a global constructive procedure for the solution of the nonlinear inverse problem considered, and we establish the uniqueness of the solution of the inverse problem.


INTRODUCTION
We study inverse spectral problems for Sturm -Liouville differential operators on a closed set of the real line (in literature it is sometimes called a time scale). Such problems often appear in natural sciences and engineering (see monographs [1,2]).
Inverse spectral problems consist in constructing operators with given spectral characteristics. For the classical differential operators on an interval inverse problems have been studied fairly completely; the main results can be found in [3][4][5][6][7][8][9][10][11][12][13][14][15]. However, differential operators defined on closed sets are essentially more difficult for investigating, and nowadays there is no inverse problem theory for this class of operators.
The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. In this paper we will study inverse problems for an important subclass of closed sets. We establish properties of spectral characteristics of Sturm -Liouville operators on closed sets and study the inverse problem of recovering the potential of the Sturm -Liouville operator from the given two spectra. The main results of the paper are Theorem 1 and Algorithm 1, where a global constructive procedure for solving the inverse problem is provided, and the uniqueness of the solution is proved.
Let us recall some notions of the time scale theory; see [1,2] for more details. Let be a closed subset of the real line; it is called sometimes the time scale. We define the so-called jump functions and 0 on as follows: ( ) = inf{ ∈ : > } for ̸ = sup , and (sup ) = sup ; 0 ( ) = sup{ ∈ : < } for ̸ = inf , and 0 (inf ) = inf . A point ∈ is called left-dense, left-isolated, right-dense and right-isolated, if 0 ( ) = , 0 ( ) < , ( ) = and ( ) > , respectively. If 0 ( ) < < ( ), then A function on is called delta-differentiable at ∈ 0 , if for any > 0 there exists a neighborhood = ( − , + ) ∩ such that for all ∈ . We will call Δ ( ) the delta-derivative of at . Example 1. If is a right-isolated point, then In particular, if = { = ℎ : ∈ Z}, then Example 2. If ∈ is a right-dense point, and is a delta-differentiable at , then In particular, if ∈ is a dense point, and is a delta-differentiable at , then is differentiable at , and Δ ( ) = ′ ( ).

DIFFERENTIAL EQUATIONS ON CLOSED SETS
Consider the Sturm -Liouville equation on : Here is the spectral parameter, ( ) ∈ is a complex-valued function. A function is called a solution of equation (1), if ∈ 2 and satisfies equation (1). The statement and the study of inverse spectral problems essentially depend on the structure of the time scale . It is necessary to choose and describe subclasses of time scales for which the inverse problem theory can be constructed adequately. In this paper we consider one of such subclasses, namely, the so-called 1-structure. More precisely, we consider the time scale of the form For 1-structure one has In particular, this yields Δ ( 1 ) = ′ ( 1 ), and consequently, Using (1) and (2) we obtain Therefore Let = 2 . It follows from (3) and (5)-(6) that where ( ) are polynomials with respect to of degree 2( + − 3), and they depend on ( 1 ), . . . , ( + −3 ). Moreover, where . Without loss of generality we assume that 1 = 0.
By the well-known arguments (see, for example [5] we obtain the following fact. where It is known (see, for example, [5]) that there exists a fundamental system of solutions of equation ( The function Φ( , ) is the solution of equation (4) satisfying the boundary conditions (9) and the jump conditions (7), i.e.

SOLUTION OF THE INVERSE PROBLEM
Let the numbers ( 2 ), . . . , ( −1 ) be known a priori. The inverse problem is formulate as follows.
Inverse problem 1. Given two spectra { } 0 , = 0, 1, construct on . In order to solve this inverse problem we will use the ideas of the method of spectral mappings [6]. Let us prove the uniqueness theorem for the solution of Inverse problem 1. For this purpose together with we consider boundary value problems˜of the same form but with the other potential˜. We agree that if a certain symbol denotes an object related to , then˜will denote an analogous object related to˜.