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

Abstract

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 Rogers 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.