# Spectral Theory of Infinite Quantum Graphs

## TL;DR

This paper explores the spectral properties of infinite quantum graphs without edge length restrictions, establishing a link with weighted discrete Laplacians and deriving new spectral results.

## Contribution

It introduces a novel connection between quantum graph spectra and discrete Laplacian spectra, leading to new self-adjointness and spectral estimates for infinite quantum graphs.

## Key findings

- Proved self-adjointness results including a Gaffney type theorem
- Derived bounds for spectra and essential spectra of quantum graphs
- Studied spectral types and lower semiboundedness

## Abstract

We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.