Oktay Sh. Mukhtarov

Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point

Abstract The purpose of this article is to extend some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundary-value problem, which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions. We construct the resolvent operator and Greens function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L 2 [a, b].

Transmission problems for the Sturm-Liouville equation involving an abstract linear operator

The goal of this paper is to study a Sturm-Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but also in one of the transmission conditions. Moreover, the equation contains an abstract linear operator. We establish such properties as isomorphism, coercive solvability and prove the discreteness of the spectrum. Finally we investigate an asymptotic behavior of the eigenvalues.

Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piece-wise continuous potential together with eigenparameter-dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular it is shown that the generalized eigen-functions form a Riesz basis of the adequate Hilbert space.

Class of Sturm–Liouville Problems with Eigenparameter Dependent Transmission Conditions

ABSTRACT The objective of this paper is the investigation of a new class Sturm–Liouville problems on two disjoint intervals when the eigenparameter appears not only in the equation but also in the boundary and transmission conditions. We concerned with the problem of justifying some spectral properties of the eigenvalues and corresponding eigenfunctions. Particularly, we examined asymptotic behavior of the eigenvalues and corresponding eigenfunctions.

Spectrum and Green’s Function of a Many-Interval Sturm–Liouville Problem

Abstract This article considers a Sturm–Liouville-type problem on a finite number disjoint intervals together with transmission conditions at the points of interaction. We introduce a new operator-theoretic formulation in such a way that the problem under consideration can be interpreted as a spectral problem for a suitable self-adjoint operator. We investigate some principal properties of eigenvalues, eigenfunctions, and resolvent operator. Particularly, by applying Green’s function method, it is shown that the problem has only point spectrum, and the set of eigenfunctions form a basis of the adequate Hilbert space.

Differential operator equations with interface conditions in modified direct sum spaces

We investigate a new type boundary value problem consisting of a differential-operator equation, eigendependent boundary conditions and two supplementary conditions so-called interface conditions. We give a characterisation of some spectral properties of the considered problem. Particularly, it is established such properties as isomorphism and coerciveness, discreteness of the spectrum and find asymptotic formulas for eigenvalues.

Lower bound estimation for eigenvalues for many interval BVP’s with eigenparameter dependent boundary conditions

The present paper deals with multi-interval Sturm-Liouville equations with eigenparameter dependent boundary-transmission conditions. Such type of problems cannot be treated with the usual techniques within the standard framework os classical Sturmian theory. It is well-known that any eigenvalue of the classical Sturm-Liouville problems can be related to its eigenfunction by the Rayleigh quotient and some useful results can be obtained from the Rayleigh quotient without solving the differential equation. For instance, it can be quite useful in estimating the eigenvalues. In this study we present a new technique for investigation some computational aspects of the eigenvalues. Particularly, we give an operator-pencil formulation of the problem and establish lower bound estimation for eigenvalues by using modified Rayleigh quotient.

Positiveness of second order differential operators with interior singularity

The main goal of this study is to provide an operator-theoretic framework for the investigation of discontinuous Sturm-Liouville problems with eigenparameter appearing in the boundary conditions. We introduce some self-adjoint compact operators in suitable Sobolev spaces such a way that the considered problem can be reduced to an operator-pencil equation. We define a new concept so-called generalized eigenfunctions and prove positiveness of operator-pencil corresponding to the considered problem.

Solvability of boundary value transmission problems by Green’s function method

In this study we give a comprehensive treatment for a class BVP’s together with transmission (impulsive, jump or interface) conditions at the one interior singular point. A self-adjoint linear operator ℛ is defined in a suitable Hilbert space ℋ such that the eigenvalues of such a problem coincide with those of ℋ and showed that it has a compact resolvent. We construct fundamental solutions and discuss some properties of spectrum. Moreover, we obtain the representation of Green’s function for the considered problem.