Spectra of discrete Schrödinger operators with primitive invertible substitution potentials

Abstract

We study the spectral properties of discrete Schrödinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to 1 as coupling constant λ tends to 0. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to λ. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to 1 as λ tends to 0.

Exact-Dimensional Property of Density of States Measure of Sturm Hamiltonian

“…The dimensional properties of its DOS have been studied in many works, see for example [28,6,7,8,26], especially the recent work [9]. We summarize the results which are related to our paper as follows: N α 1 ,λ is exact-dimensional and Girand [12] and Mei [23] considered the frequency α with eventually periodic continued fraction expansion. In both papers they showed that lim λ→0 d(α, λ) = 1 and N α,λ is exact dimensional for small λ.…”

On sums of nearly affine Cantor sets

“…log ∂k Notice that Lemma 3.5 implies also that for come constantĈ > 0 we have Therefore one can choose smallness of the C 2 norms of {g i } i=1,...,m in (8) so that for some δ * > 0 and all large enough values of n ∈ N we have d dλ φ ω,τ (λ) ≥ nδ * n s=1 l (s) (26) for any ω, τ ∈ Ω with |ω ∧ τ | = n. In particular, if ω, τ ∈ Ω 1 then (26) together with (19) imply that d dλ φ ω,τ (λ) ≥ nδ * m −βn = nδ * m −β|ω∧τ | , which implies (20) and hence verifies the assumption (13). Finally, the Shannon-McMillan-Breiman Theorem implies that…”

Quantum and spectral properties of the Labyrinth model

“…Theorem 2.2 ([2], see also Theorem 4.1 from [26]). The set Λ λ is a compact locally maximal T s -invariant transitive hyperbolic subset of S λ , and the periodic points of T s form a dense subset of Λ λ .…”