Dynamical Bounds for Sturmian Schrödinger Operators

Abstract

The Fibonacci Hamiltonian, that is a Schrödinger 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 in [10] and used it to exhibit 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 counterexample, we exhibit a pathological irrational number which does 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.

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

“…The proof was based on methods from [25], which were since found to be flawed [3]. We believe that the extension can be proved by combining our methods with the ones in [3], but we leave this task to future work.…”

Mathematics of Aperiodic Order

“…The following result from [68] gives dynamical lower bounds for all values of λ and p, provided α has bounded continued fraction coefficients. The following result from [135] gives dynamical upper bounds in the large coupling regime. Theorem 6.5.…”

“…The transport exponents in the Sturmian case were studied in the papers [40,58,66,68,135]. The following result from [68] gives dynamical lower bounds for all values of λ and p, provided α has bounded continued fraction coefficients.…”

Transport exponents of Sturmian Hamiltonians

“…Moreover, we have the general boundsβ + (p) ≤ β + (p) for every p > 0; see (the proof of) [15,Lemma 7.2]. In particular, for the operator H λ,α,ω found in the theorem, we haveα [29], which is the basis for the assertion in [29,Theorem 2], claims that there exists α such that for every λ > 20, we haveα + u = 1 for the operator H λ,α,0 . What we add here is the extension throughout the entire range of moments p > 0 and couplings λ > 0, the consideration of both time-averaged and non-time-averaged quantities, and the genericity of the set of frequencies for which such a statement can be shown.…”

“…Theorem 1.1 at first glance looks like merely a slight improvement over [29,Corollary 1] (which implies that lim sup λ→∞α + u ·log λ ≤ π 2 6 log 2 ), but it gives a bound that appears to be a good candidate for a sharp bound (or even exact asymptotics). Moreover, as we will discuss below, the proofs in [29] have several gaps, so that most of the main results of [29], including [29,Corollary 1], are actually not completely proved there. For some of the results there it is even doubtful whether they are true as stated.…”