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

## DOI:

https://doi.org/10.26577/jmmcs-2018-1-486## Keywords:

oriented graph, vertices of graph, Kifchhoff condition, vibrations of elastic networks, Green’s function of Dirichlet problem, extension by eigenfunctions## 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.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*97*(1), 67–90. https://doi.org/10.26577/jmmcs-2018-1-486