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 introduce a class of Jacobi matrices with sparse potential in the sense that the perturbation of the Laplacian consists of a (direct) sum of fixed offdiagonal 2 × 2 matrices placed at sites whose distances from one another grow exponentially. The essential spectrum is proved to be the set [−2, 2]. The model may be formulated in terms of angular variables, the Prüfer angles, in terms of which is defined a nonlinear dynamical system with two control parameters, p-the 'impurity strength'-and β-the 'sparseness'. A method to study the spectrum of these matrices which exploits the existence, for fixed energy, of a continuous asymptotic distribution function (a.d.f.) of the Prüfer angles is introduced. This is in contrast to the method introduced by Pearson (1978 Commun. Math. Phys. 60 13-36), which exploits uniform distribution in energy. With the help of metric theorems of ergodic theory and ideas of Zlatoš (2004 J. Funct. Anal. 207 216-52), we are able to use this framework in order to establish a singular continuous spectrum on an interval of positive Lebesgue measure, under suitable conditions on β and p. In the complementary set of parameters there is a (dense) pure point spectrum, and thus a spectral transition takes place, if a continuous a.d.f is assumed. We provide numerical evidence for this assumption.

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