# Laplacians on periodic graphs with guides

## TL;DR

This paper studies how adding guides to periodic graphs affects the Laplacian spectrum, revealing new guided bands and providing estimates for their positions and lengths based on graph geometry.

## Contribution

It introduces the concept of guided spectrum in perturbed periodic graphs and analyzes their properties, including asymptotics and possible configurations.

## Key findings

- Guided spectrum consists of a finite union of bands added to the original spectrum.
- Positions and lengths of guided bands are estimated using geometric parameters.
- The number and properties of guided bands can be highly variable depending on the graph structure.

## Abstract

We consider Laplace operators on periodic discrete graphs perturbed by guides, i.e., graphs which are periodic in some directions and finite in other ones. The spectrum of the Laplacian on the unperturbed graph is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed Laplacian consists of the unperturbed one plus the additional so-called guided spectrum which is a union of a finite number of bands. We estimate the position of the guided bands and their length in terms of geometric parameters of the graph. We also determine the asymptotics of the guided bands for guides with large multiplicity of edges. Moreover, we show that the possible number of guided bands, their length and position can be rather arbitrary for some specific periodic graphs with guides.