We study the problem of site recurrence of discrete time nearest neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks our discussion concerns two notions of recurrence, one given by a monitoring procedure [9,13], another in terms of Pólya numbers [22], and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks are related by an additive interference term in a simple way. Based on a previous result of positive recurrence we describe an open quantum version of Kac's lemma for the expected return time to a site.

We study spectral properties of some discrete Dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions. As observed in the corresponding Schrödinger operators, we determine the Hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous. C

Existence of generic sets of self-adjoint operators, related to correlation dimensions of spectral measures, is investigated in separable Hilbert spaces. Typical results say that, given an orthonormal basis, the set of operators whose corresponding spectral measures are both 0-lower and 1-upper correlation dimensional is generic. The proofs rely on details of the relations among Fourier transform of spectral measures and Hausdorff and packing measures on the real line. Then such results are naturally combined with the Wonderland theorem. Applications are to classes of discrete one-dimensional Schrödinger operators and general (bounded) self-adjoint operators as well. Physical consequences include a proof of exotic dynamical behavior of singular continuous spectrum in some settings.

We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i.e. nonrandom, block-Jacobi matrices may be determined with any degree of precision, improving a result of Zlatoš [Andrej Zlatoš, Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004) 216-252].

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, [Formula: see text], in a product space whose alphabet is a perfect Polish metric space (thus, uncountable). More specifically, we show that the set of invariant measures with upper Hausdorff dimension equal to zero and lower packing dimension equal to infinity is a dense [Formula: see text] subset of [Formula: see text], the space of [Formula: see text]-invariant measures endowed with the weak topology. We also show that the set of invariant measures with upper rate of recurrence equal to infinity and lower rate of recurrence equal to zero is a [Formula: see text] subset of [Formula: see text]. Furthermore, we show that the set of invariant measures with upper quantitative waiting time indicator equal to infinity and lower quantitative waiting time indicator equal to zero is residual in [Formula: see text].