R. del Rio

Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization

Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recently within spectral theory by Last [22] (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one perturbations (and discussed in [7,11,33]) from this point of view. A Borel measure μ is said to have exact dimension α ∈ [0, 1] if and only if μ(S) = 0 if S has dimension β < α and if μ is supported by a set of dimension α. If 0 < α < 1, such a measure is, of necessity, singular continuous. But, there are also singular continuous measures of exact dimension 0 and 1 which are “particularly close” to point and a.c. measures, respectively. Indeed, as we’ll explain, we know of “explicit” Schrödinger operators with exact dimension 0 and 1, but, while they presumably exist, we don’t know of any with dimension α ∈ (0, 1). While we’re interested in the abstract theory of rank one perturbations, we’re especially interested in those rank one perturbations obtained by taking a random Jacobi matrix and making a Baire generic perturbation of the potential at a single point. It is a disturbing fact that the strict localization (dense point spectrum with ‖xe−itHδ0‖2 = (e−itHδ0, x2e−itHδ0) bounded in t), that holds a.e. for the random case, can be destroyed by arbitrarily small local perturbations [7,11]. We’ll ameliorate this discovery in the present paper in three ways: First, we’ll see that, in this case, the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous. Second, we’ll show that not

Operators with singular continuous spectrum. II. Rank one operators

For an operator,A, with cyclic vector ϕ, we studyA+λP, whereP is the rank one projection onto multiples of ϕ. If [α,β] ⊂ spec (A) andA has no a.c. spectrum, we prove thatA+λP has purely singular continuous spectrum on (α,β) for a denseGδ of λs.

Singular continuous spectrum is generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense G delta.

Inverse Spectral Results for AKNS Systems with Partial Information on the Potentials

For the AKNS operator on L2([0,1],C2) it is well known that the data of two spectra uniquely determine the corresponding potential ϕ a.e. on [0,1] (Borgs type Theorem). We prove that, in the case where ϕ is a-priori known on [a,1], then only a part (depending on a) of two spectra determine ϕ on [0,1]. Our results include generalizations for Dirac systems of classical results obtained by Hochstadt and Lieberman for the Sturm–Liouville case, where they showed that half of the potential and one spectrum determine all the potential functions. An important ingredient in our strategy is the link between the rate of growth of an entire function and the distribution of its zeros.