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

v3: 12 pages, 2 figures; to appear in the Journal of Mathematical Physics in May or June 2016/ JMP 2016

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Cited by in corpus (6)

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• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• Regularized Laplacian determinants of self-similar fractals
• From non-symmetric particle systems to non-linear PDEs on fractals
• Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs