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

2015/09/30 by Joe P. Chen, Alexander Teplyaev
Materials Science · Mathematics · Physics and Astronomy · #Absolute continuity #Continuous spectrum #Integer (computer science) #Laplace operator #Laplacian matrix #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Spectral theory #Spectrum (functional analysis) #Type (biology) #math-ph #math.DS #math.MP #math.SP #msc:28A80 #msc:37F50 #msc:47A10 #msc:60J35 #msc:81Q12 #msc:81Q35

paper · pdf · doi:10.1063/1.4949471

published as J. Math. Phys. 57, 052104 (2016) · v3: 12 pages, 2 figures; to appear in the Journal of Mathematical Physics in May or June 2016/ JMP 2016

openalex publication_date 2016/05/01 · arxiv created 2016/05/23 · arxiv updated 2016/05/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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.

Citations

• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Spontaneous emission from a fractal vacuum
• Weak uncertainty principle for fractals, graphs and metric measure spaces
• Heat kernels and zeta functions on fractals
• Topological properties of Fibonacci quasicrystals : A scattering analysis of Chern numbers