ON BASIS PROPERTY OF ROOT FUNCTIONS FOR A CLASS OF THE SECOND ORDER DIFFERENTIAL OPERATORS

It is well known that the Sturmian theory is an important tool in solving numerous problems of mathematical physics. Usually, eigenvalue parameter appears linearly only in the differential equation of the classic Sturm–Liouville problems. However, in mathematical physics there are also problems, which contain eigenvalue parameter not only in differential equation, but also in the boundary conditions. In this paper, we consider a Sturm–Liouville equation with the eigenparameter dependent boundary condition and with transmission conditions at two points of discontinuity. The aim of this paper is to investigate the completeness, minimality and basis properties of rootfunctions for the considered boundary value problem.


Introduction
In this work, we consider ( ) : for        . In [16], the asymptotic formulas for the eigenvalues and eigenfunctions of problem (1)-(7) are obtained. Spectral problems for Sturm-Liouville equations with the eigenparameter dependent boundary conditions are of particular interest due to physical applications and are examined in [2,6,9,10,17]. To this end, the method of separation of variables is applied to solve the corresponding partial differential equation when the boundary conditions contain a directional derivative. Problems on eigenvalue for the second order equation with spectral parameter in the boundary conditions are considered in [5,7,8,[11][12][13][14][15]18]. The corresponding problems led to the eigenvalue problem for a linear operator acting on the space where N is N  dimensional Euclidean space of complex numbers. In [9], for distinct cases, it is shown that the eigenfunctions of the spectral problem formed a defect basis in   2 0,1
The goal of this work is to investigate the problem of completeness, minimality and basis property of the eigenfunctions of boundary value problem (1)- (7). In this study, we introduce a special inner product in a special Hilbert space and construct a linear operator A in the space such that problem (1)- (7) can be interpreted as the eigenvalue problem for A .

Operator Theoretic Formulation of the Problem
In this section, we introduce a special inner product in the Hilbert Space (,) denotes the inner product in 2 [ 1,1].
In this case, the operator A is not selfadjoint in the space H . Therefore, we introduce the operator J as 0 =, 0 I is the identity operator in . H The operator J is selfadjoint and invertible. In this case, boundary value problem (1)- (7) is equivalent to the eigenvalue problem for the operator pencil Pu x u x whose one element is omitted forms a complete and minimal system in 2 ( ) = [ 1,1]. P H L  Hence, the eigenfunctions nn  0 n is an arbitrary nonnegative integer) of boundary value problem (1)-(7) form complete and minimal system in 2 [ 1,1].
L  This completes the proof of Corollary 1. HL  This completes the proof of Theorem 2.

Results and Discussion
The paper is devoted to one class of the Sturm-Liouville operators with the eigenparameterdependent boundary conditions and the transmission conditions. A new operator A associated with the problem is established, some spectral properties of this operator is examined in an appropriate space H and basisness of its eigenfunctions is discussed.