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 ${\mathbb{Z}}_{+}$. Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever $p\mathrm{\ne }\frac{1}{2}$. 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.