Spectral decimation of a self-similar version of almost Mathieu-type operators

TL;DR

This paper introduces self-similar versions of almost Mathieu operators using self-similar Laplacians, establishing their spectral properties via spectral decimation, and deriving explicit formulas for their integrated density of states.

Contribution

It defines a new class of self-similar almost Mathieu operators and connects their spectra to self-similar Laplacians through spectral decimation, extending analysis on fractals.

Findings

Spectra can be fully described via spectral decimation.

Emergence of singularly continuous spectra for certain parameters.

Explicit formula for the integrated density of states.

Abstract

We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian, and includes the classical almost Mathieu operators as a particular case. Our main result establishes that the spectra of these self-similar almost Mathieu operators can be completely described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. In addition, the self-similar structure of our model provides a natural finite graph approximation model. This approximation is not only helpful in executing the numerical simulation, but is also useful in finding the spectral decimation function via Schur complement computations of given finite-dimensional matrices. The self-similar Laplacians used in our model were considered recently by Chen and Teplyaev who proved the emergence of singularly continuous spectra for specific parameters. We use this result to arrive at similar conclusions in the context of the self-similar almost Mathieu operators. Finally, we derive an explicit formula of the integrated density of states of the self-similar almost Mathieu operators as the weighted pre-images of the balanced invariant measure on a specific Julia set.