Pseudodifferential operators on periodic graphs

TL;DR

This paper investigates the Fredholm properties of pseudodifferential operators on periodic infinite metric graphs, combining local Fourier and Mellin pseudodifferential operator analysis to understand operator behavior.

Contribution

It introduces a framework for analyzing Fredholm properties of operators acting as Fourier pseudodifferential operators on edges and matrix Mellin operators near vertices of periodic graphs.

Findings

Fredholm criteria established for pseudodifferential operators on periodic graphs

Application to singular integral operators on graphs

Extension of operator theory to graph-based structures

Abstract

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph $\Gamma$ which is periodic with respect to the action of the group ${\mathbb Z}^n$. The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class $OPS^0$ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of $\Gamma$. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.