Green’s function of the Dirichlet problem for the differential operator on a star-shaped graph at m (Journal of Mathematics Mechanics and Computer Science, 2019)
In this paper, we investigate a system of second-order differential equations, which is a model of oscillatory systems with a rod structure. Problems for differential operators on graphs are currently being actively studied by mathematicians and have applications in quantum mechanics, organic chemistry, nanotechnology, the theory of waveguides and other fields of natural science. A graph is a structure consisting of "abstract" segments and vertices whose adjoining to each other is described by a certain relation. To define an operator on a given graph, it is necessary to select a set of boundary vertices. Vertices that are not boundary are called internal vertices. The differential operator on a given graph is determined not only by given differential expressions on arcs, but also by conditions of the Kirchhoff type at the internal vertices of the graph. This article solved the Dirichlet problem for a differential operator on a star graph. We used the standard gluing conditions at inner vertices and Dirichlet boundary conditions at the boundary vertices. Also in this paper, the subtraction of Green's function of a differential operator on a star shaped graph is presented. Questions from spectral theory, such as the construction of the Green function and the expansion in eigenfunctions for models from connected rods, have been little studied. Spectral analysis of differential operators on graphs is the main mathematical apparatus in solving modern problems of quantum mechanics.
Keywords: oriented graph; vertices of graph; Kifchhoff condition; vibrations of elastic networks; Dirichlet problem; extension by eigenfunctions