Residual Decomposition of the Green's Function of the Dirichlet problem for a Differential Operator on a star-graph for m=2
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Abstract
In this paper, we have researched 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, waveguide theory, and other areas of natural science. A graph is a structure consisting of "abstract" segments and vertexes, the adjacency of which to each other is described by some relation. To define an operator on a given graph, it is necessary to select a set of boundary vertexes. Vertexes that are not boundary are called internal vertexes. A differential operator on a given graph is determined not only by given differential expressions on edges, but also by Kirchhoff-type conditions at internal vertexes of the graph. In this article, the Dirichlet problem for a differential operator on a star graph is solved. We have used standard gluing conditions at internal vertexes and Dirichlet boundary conditions at boundary vertexes. Also in this paper, the Green's function of a differential operator on a star graph is presented. Questions from the spectral theory, such as the construction of the Green's function and the expansion in terms of eigenfunctions for models of connected rods, have been little studied. Spectral analysis of differential operators on graphs is the main mathematical tool for solving modern problems of quantum mechanics
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