K. Aydemir

Generalized eigenfunctions of one Sturm-Liouville system with symmetric jump conditions

The aim of this study is to investigate one discontinuous Sturm-Liouville problem with eigenparameter appearing in the boundary-transmission conditions. We esthabilish some spectral properties for the considered problem in suitable Sobolev spaces. The main result of this study state that the system of generalized eigenfunctions form a Riesz basis of the appropriate Hilbert space.

New Type of Sturm-Liouville Problems in Associated Hilbert Spaces

We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

Riesz basis property of weak eigenfunctions for boundary-value problem with discontinuities at two interior points

We investigate one discontinuous boundary value problem which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. We establish some spectral properties of the considered problem. For the problem under consideration we define a new concept so-called weak eigenfunctions which is an extension of a classical eigenfunction and prove that the system of weak eigen-functions form a Riesz basis of the appropriate Hilbert space for the modified Lebesgue space.

Spectrum of one BVP with discontinuities and spectral parameter in the boundary conditions

The aim this of paper is to investigate the spectral problem for the equation −(pu′)′(x) + q(x)u(x) = λu(x), under eigen-dependent boundary conditions and supplementary transmission conditions at finite number interior points. By modifying some techniques of classical Sturm-Liouville theory and suggesting own approaches we esthabilish some properties of the eigenvalues and eigenfunction.

The resolvent operator of one BVP with interior singularities

In this work we consider a class of Sturm-Liouville problems with eigenparameter dependent boundary and transmission conditions. By suggesting an own approaches we have presented a formula for Green’s function. We construct a new Hilbert space and linear selfadjoint operator in it for operator realization of the problem, and state the main spectral properties of this operator.

Asymptotics eigenvalues for many-interval Sturm-Liouville problems

In this work we consider a Sturm-liouville equation on finite number disjoint intervals together with transmission conditions at the points of interaction. Moreover, the eigenvalues parameter appears linearly in the boundary conditions. We investigate some properties of eigenvalues and eigenfunctions of the considered problem and derive asymptotic formulas for the eigenvalues.