Green's Function and Carleman's Formula for Transmission Problems

Abstract

This work is devoted to the study of some spectral properties of two-interval Sturm-Liouville problems of new type together with transmission condition at the interaction point. Here, the interaction point is viewed as a right endpoint of the left interval and a left endpoint of the right interval. In fact, we are studying two different Sturm-Liouville equations for two unknown solutions; one is defined on the left interval Xi(l) = (-1, 0), and the other is defined on the right interval Xi(r) = (0, 1), having a common endpoint x = 0, so-called interaction point, on which additional transmission conditions are imposed. Using our own approach, we establish such important spectral properties, as the expansion of the Green's function in a series of eigenfunctions, Parseval equality and Carleman's formula.