Green’s function of the Dirichlet problem for the differential operator on a star-shaped graph - Kanguzhin, B E. (Journal of Mathematics, Mechanics and Computer Science, 2018)
Differential operators on graphs often arise in mathematics and different fields of science such as mechanics, physics, organic chemistry, nanotechnology. In this paper we deduced the Green function of the Dirichlet problem for a differential operator on a star-shaped graph.We study the differential operator with standard matching conditions in the internal vertices and the Dirichlet boundary conditions at boundary vertices. In this paper, we investigate a system of second-order differential equations, which is a model of vibrational systems with a rod structure. Problems for differential operators on graphs are now actively studied by mathematicians and have applications in quantum mechanics, organic chemistry, nanotechnology, waveguide theory and other fields of natural science. In this paper we derive the Green function of the Dirichlet problem for a differential operator on a starlike graph. A significant difficulty is the construction of the Green’s function on geometric graphs for values of independent variables close to the vertices of the graph. We used standard gluing conditions in internal vertices and Dirichlet boundary conditions at boundary vertices. A constructive scheme for constructing the Green’s function of the boundary value problem for the Sturm-Liouville equation is proposed. The existence of a decomposition of an arbitrary function defined on a graph with respect to eigenfunctions is proved. Questions from the spectral theory, like the construction of the Green’s function and the expansion in eigenfunctions for models from connected rods, have so far been little studied. Spectral analysis of differential operators on geometric graphs is the basic mathematical apparatus in solving modern problems of quantum mechanics.