Green`s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space

Authors : Oktay MUKHTAROV, Kadriye AYDEMİR, Hayati OLĞAR

Pages : 117-120

View : 44 | Download : 7

Publication Date : 2018-12-29

Article Type : Conference Paper

Abstract :The theme of the development of the theory and applications of Green`s Functions is skilfully used to motivate and connect clear accounts of the theory of distributions, Fourier series and transforms, Hilbert spaces, linear integral equations. In this work we analyze the Green`s functions of boundary value problems defined on two interval and associated with Schrodinger operators with interaction conditions. We have constructed some special eigensolutions of this problem and presented a formula and the existence condition of Green`s function in terms of the general solution of a corresponding homogeneous equation. We have obtained the relation between two Green`s functions of two nonhomogeneous problems. It allows us to find Green`s function for the same equation but with different additional conditions. These problems include the cases in which the boundary has two, one or none vertices. In each case, the Green`s functions, the eigenvalues and the eigenfunctions are given in terms of asymptotic formulas. A preliminary study of two-point regular boundary value problems with additional transmission conditions was developed by the authors of this study  under the denomination of two-point transmission boundary value problems. In each case, it is essential to describe the solutions of the Schrodinger equation on the interior nodes of the path. As an consequence of this property, we can characterize those boundary value problems that are regular and then we obtain their corresponding Green`s function, as well as the eigenvalues and the eigenfunctions for the regular case.
Keywords : Schrodinger operators, Green Functions, Transmission conditions

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