Spectral Dimension for $\beta$-almost periodic singular Jacobi operators and the extended Harper's model

TL;DR

This paper investigates the fractal spectral properties of singular Jacobi operators, establishing bounds and criteria for spectral dimensionality, with applications to the extended Harper's model and quantum dynamics.

Contribution

It provides new quantitative bounds and an arithmetic criterion for spectral dimensionality in quasiperiodic Jacobi operators, extending understanding of spectral and dynamical properties.

Findings

Quantitative lower spectral bounds for operators with strong repetition

Arithmetic criterion for full spectral dimensionality in quasiperiodic operators

Results on spectral dimensions and quantum dynamical exponents for the extended Harper's model

Abstract

We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper's model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.