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

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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