Dimensional Hausdorff Properties of Singular Continuous Spectra

Abstract

We present an extension of the Gilbert-Pearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of one-dimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze these properties for several examples having the singular-continuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.PACS numbers: 02.30. Sa, 71.23.An, 72.15.Rn Singular continuous spectra have been extensively studied recently. Our interest here is in the classification and decomposition of such spectra with respect to dimensional Hausdorff measures. The measure-theoretical aspect of this point of view goes back to Rogers and Taylor [1], and it has been studied recently within spectral theory by Last [2] and by del Rio et al. [3] who have shown that the singular-continuous spectrum which is produced by localized rank-one perturbations of Anderson-model Hamiltonians in the localized regime [4] must be purely zero dimensional-in the sense that the associated spectral measures are supported on a set of zero Hausdorff dimension.The main purpose of this paper is to report a general method for spectral analysis of one-dimensional Schrödinger operators from this point of view. It is a natural extension of the Gilbert-Pearson theory of subordinacy [5,6], and it allows us to analyze the dimensional Hausdorff properties for a number of examples with the singular-continuous spectrum. Below we describe the main ideas of our study and some of the main results. Mathematically complete proofs of these results will be given elsewhere [7].Most of our discussion will be restricted to onedimensional discrete (tight-binding) Schrödinger operators of the form ͑Hc͒ ͑n͒ c͑n 1 1͒ 1 c͑n 2 1͒ 1 V͑n͒c͑n͒ .We shall consider two kinds of such operators: "line" operators acting on ᐉ 2 ‫͒ޚ͑‬ ͑2`, n ,`͒, and "half-line" operators acting on ᐉ 2 ‫ޚ͑‬ 1 ͒ ͑n . 0͒, which are considered with a phase boundary condition of the formwhere 2p͞2 , u , p͞2. Before formulating our main result, which would require some definitions, we would like to describe some of its applications. We stress at this point that the dimensional Hausdorff properties which we study are those which are associated with the spectral measures of the corresponding operators. The spectra themselves, as sets, are closed sets, and their dimensions may be larger than those which are associated with the spectral measures. A description of the precise spectral-theoretic scheme which underlies our study is given below.We start with a somewhat artificial example of half-line operators with sparse barrier potentials. More specifically, we consider potentials which vanish for all n's outside a sparse (fastly growing) sequence of points ͕L n ͖ǹ 1 where jV ͑L n ͒j !`as n !`. Simon and Spencer [8] have shown that the Schrödinger operators corresponding to such potentials have no absolutely continuous spectrum, and Gordon [9] has shown that if the jV ͑L n ͒j's grow suffi...

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