Singularly continuous spectrum of a self-similar Laplacian on the half-line

TL;DR

This paper demonstrates that a self-similar Laplacian on the half-line exhibits singularly continuous spectrum for most parameter values, providing a simple model for understanding such spectral types in self-similar structures.

Contribution

It introduces a self-similar Laplacian model on the half-line and proves its spectrum is singularly continuous when p ≠ 1/2, advancing understanding of spectral types in self-similar systems.

Findings

Spectral decimation method applied to the Laplacian.

Singularly continuous spectrum occurs for p ≠ 1/2.

Model serves as a toy example for complex self-similar spectra.

Abstract

We investigate the spectrum of the self-similar Laplacian, which generates the so-called "$pq$ random walk" on the integer half-line $\mathbb{Z}_+$. Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever $p\neq \frac{1}{2}$. This serves as a toy model for generating singularly continuous spectrum, which can be generalized to more complicated settings. We hope it will provide more insight into Fibonacci and other weakly self-similar models.