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

Abstract

We investigate the spectrum of the self-similar Laplacian, which generates the so-called “pq random walk” on the integer half-line ℤ+. Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever p≠12. 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-type and other weakly self-similar models.