Dynamical bounds for Sturmian Schrödinger operators

Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian

“…, is replaced by a general irrational α ∈ (0, 1) in ( 35) and ( 36). The one-dimensional case has been investigated to a great extent (see, e.g., [3,20,21,30,38,41,42,45,59], among many others), thereby opening the door for a study of separable models in higher dimensions based on these one-dimensional operators.…”

“…Let d ≥ 1 be an integer and assume that for 1 ≤ j ≤ d, we have bounded maps V j : Z → R. Consider the associated Schrödinger operators on 2 (Z), (41) […”

On- And Off-Diagonal Sturmian Operators: Dynamic and Spectral Dimension

“…Thus, the study of Sturmian operators was further motivated by the interest in electronic spectra and transport properties of one-dimensional quasicrystals. It is shown that those operators give rise to anomalous transport for a large class of irrational number [DT1,M1,M2]. Moreover, it exhibits a number of interesting phenomena, such as Cantor spectrum of Lebesgue measure zero [S1,S3,BIST] and purely singular continuous spectral measure [DL, DKL, BIST].…”

“…The proof is a direct consequence of properties of the functions x k (E) which are proper and continuous as polynomials in E and the distorsion theorem of Koebe. See [DT1,M2,M1] for details.…”

Transport exponents of Sturmian Hamiltonians

“…), 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.…”