# Quantum graphs on radially symmetric antitrees

## TL;DR

This paper analyzes the spectral properties of Kirchhoff Laplacians on radially symmetric antitrees, revealing their decomposition, deficiency indices, self-adjoint extensions, and spectral types, with criteria for various spectral features.

## Contribution

It provides a detailed spectral analysis of Kirchhoff Laplacians on radially symmetric antitrees, including explicit descriptions of self-adjoint extensions and conditions for spectral properties.

## Key findings

- Deficiency indices are at most one, equal to one for finite volume antitrees.
- Explicit description of all self-adjoint extensions, including Friedrichs extension.
- Criteria for discreteness, trace class, positivity, and spectral gaps; rare occurrence of absolutely continuous spectrum.

## Abstract

We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,\infty)$.