Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

Abstract

Abstract. We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely α-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

“…As mentioned above, this also bounds the box counting dimension from below by general principles. Spectral Hausdorff continuity results for the Fibonacci Hamiltonian were shown in [6,8,19,22]. The best lower bound that has been obtained in this way can be found in [22] and it reads 5 log φ ≈ 0.0126 2 Notice that there is a typo in [22].…”

“…For θ = 0, the best bound is contained in [22] and it has (6) with f # ≈ 1.83157 replaced by (5), that is, ≈ 0.00188. For other values of θ, the best bound can be found in [8] and the constant in this bound is even smaller.…”

“…There are two main previous approaches to quantum dynamical lower bounds for the Fibonacci model. The first is based on spectral continuity and the papers [6,8,19,22] contain results obtained in this way. For θ = 0, the best bound is contained in [22] and it has (6) with f # ≈ 1.83157 replaced by (5), that is, ≈ 0.00188.…”

“…For θ = 0, the paper [11] has (6) with f # replaced by 1/6. It is possible to treat general θ along the same lines using [10], but the dynamical lower bound has a somewhat smaller constant in the general case.…”

Jacobi Operators and Completely Integrable Nonlinear Lattices

“…The section on almost periodic operators summarizes some results from [21] and [122]. There have recently been some efforts to prove uniform spectral results (i.e, for all operators in the hull), see, for example, [51], [52] and the references therein. In this respect, it should be mentioned that the absolutely continuous spectrum is constant for all operators in the hull ( [163], Theorem 1.5).…”