Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian

TL;DR

This paper demonstrates that the density of states measure for the square Fibonacci Hamiltonian is absolutely continuous for most small coupling constants, using new results on convolutions of measures in hyperbolic dynamics.

Contribution

It introduces a novel result on the absolute continuity of convolutions of measures in hyperbolic dynamics and applies it to the spectral analysis of the square Fibonacci Hamiltonian.

Findings

Density of states measure is absolutely continuous for almost all small coupling constants.

New theoretical result on convolutions of measures in hyperbolic dynamics.

Application of measure convolution results to quantum Hamiltonian spectral properties.

Abstract

We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.