The fractal dimension of the spectrum of quasiperiodical schrodinger operators

TL;DR

This paper investigates the fractal and box dimensions of the spectrum of quasiperiodic Schrödinger operators with Sturmian potentials, establishing bounds linked to the properties of irrational numbers, with implications for dynamical systems.

Contribution

It provides new lower bounds on the spectrum's fractal dimension for all irrational numbers and improves these bounds for almost all irrationals, connecting spectral properties to number theory.

Findings

Established a general lower bound for the box dimension of the spectrum.

Improved the lower bound for almost all irrational numbers.

Linked spectral dimension properties to dynamical implications.

Abstract

We study the fractal dimension of the spectrum of a quasiperiodical Schrodinger operator associated to a sturmian potential. We consider potential defined with irrationnal number verifying a generic diophantine condition. We recall how shape and box dimension of the spectrum is linked to the irrational number properties. In the first place, we give general lower bound of the box dimension of the spectrum, true for all irrational numbers. In the second place, we improve this lower bound for almost all irrational numbers. We finally recall dynamical implication of the first bound.