Dynamical Bounds for Sturmian Schr\"{o}dinger Operators

TL;DR

This paper extends dynamical bounds for Sturmian Schrödinger operators, demonstrating anomalous transport for a broad class of irrational numbers under certain conditions and establishing spectral dimension bounds.

Contribution

It generalizes existing dynamical upper bounds to a large family of Sturmian operators and introduces new spectral dimension estimates.

Findings

Anomalous transport occurs for operators with generic Diophantine irrational numbers.

Counterexamples show ballistic bounds for certain pathological irrationals.

A global lower bound for the spectrum's box counting dimension is established.

Abstract

The Fibonacci Hamiltonian, that is a Schr\"{o}dinger operator associated to a quasiperiodical sturmian potential with respect to the golden mean has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method and find a non trivial dynamical upper bound for this model. In this paper, we use this method to generalize to a large family of Sturmian operators dynamical upper bounds and show at sufficently large coupling anomalous transport for operators associated to irrational number with a generic diophantine condition. As a counter example, we exhibit a pathological irrational number which do not verify this condition and show its associated dynamic exponent only has ballistic bound. Moreover, we establish a global lower bound for the lower box counting dimension of the spectrum that is used to obtain a dynamical lower bound for bounded density irrational numbers.