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Chen, Joe P.; Teplyaev, Alexander

Singularly continuous spectrum of a self-similar Laplacian on the half-line. (English) Zbl 1341.60066

J. Math. Phys. 57, No. 5, 052104, 10 p. (2016).
Summary: 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 a 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.{
©2016 American Institute of Physics}

Cited in 11 Documents

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60G18 Self-similar stochastic processes
60G50 Sums of independent random variables; random walks
35C06 Self-similar solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

self-similar Laplacian; singularly continuous spectrum; random walk
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References:

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