Singular spectrum for radial trees

TL;DR

This paper demonstrates that the absolutely continuous spectrum for the Laplacian on radial trees is uncommon, with unbounded edges leading to purely singular spectrum and generic radial trees exhibiting purely singular continuous spectrum.

Contribution

It provides new results showing the rarity of absolutely continuous spectrum and characterizes the spectral types for radial trees with unbounded edges.

Findings

Unbounded edges in metric trees lead to purely singular spectrum.

Generically, radial trees have purely singular continuous spectrum.

Absolutely continuous spectrum is a rare event for the Laplacian on radial trees.

Abstract

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that generically (in the sense of Baire) radial trees have purely singular continuous spectrum.