Green’s function of the Dirichlet problem for the differential operator on a star-shaped graph

Abstract

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.