Green Operators of Networks with a new vertex

TL;DR

This paper studies how adding a new vertex to a network affects elliptic Schr"{o}dinger operators and their Green functions, providing a way to compute the new Green function from the original network's Green function.

Contribution

It introduces a perturbation approach for elliptic Schr"{o}dinger operators on networks with an added vertex, linking the new Green function to the original.

Findings

Green functions on modified networks can be expressed via original network's Green functions.

Perturbation methods simplify the computation of operators after network modifications.

The approach applies to elliptic Schr"{o}dinger operators on networks with added vertices.

Abstract

Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schr\"{o}dinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schr\"{o}dinger operators on the initial network. Therefore, the Green function on the new network can be computed in terms of the Green function of the original network.