# Absolutely continuous spectrum for Laplacians on radial metric trees and periodicity

## TL;DR

This paper investigates the spectral properties of Laplacians on radial metric trees, showing that absolutely continuous spectrum implies eventual periodicity of geometric and coupling data, with some non-periodic exceptions.

## Contribution

It establishes a link between absolutely continuous spectrum and eventual periodicity in radial trees, extending understanding of spectral theory on such structures.

## Key findings

- Absolutely continuous spectrum implies eventual periodicity of geometric data.
- Examples of non-periodic couplings with absolutely continuous spectrum are provided.
- Conditions for the presence of absolutely continuous spectrum are characterized.

## Abstract

On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $\delta$- and weighted $\delta'$-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectrum.