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

Özet

The theme of the development of the theory and applications of Green’s Functions is skilfully usedto 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 twointerval and associated with Schrodinger operators with interaction conditions. We have constructed some specialeigensolutions of this problem and presented a formula and the existence condition of Green’s function in terms ofthe general solution of a corresponding homogeneous equation. We have obtained the relation between two Green’sfunctions of two nonhomogeneous problems. It allows us to find Green’s function for the same equation but withdifferent 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 asymptoticformulas. 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 valueproblems. In each case, it is essential to describe the solutions of the Schrodinger equation on the interior nodes ofthe path. As an consequence of this property, we can characterize those boundary value problems that are regularand then we obtain their corresponding Green’s function, as well as the eigenvalues and the eigenfunctions for theregular case.