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

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 not equal 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-type and other weakly self-similar models. Published by AIP Publishing.