The paper under review deals with discrete one-dimensional Schrödinger operators, i.e., operators in \(\ell^2(\mathbb{Z})\) of the form \[ H := \Delta + V, \] where \(\Delta\) is the discrete Laplacian and \(V\) is the multiplication operator by the potential \(V\colon \mathbb{Z}\to \mathbb{R}\), which is dynamically defined in the following sense: given a dynamical system \((\mathbb{X},S)\) (i.e.,\(\mathbb{X}\) is a compact metric space and \(S\) is a homeomorphism on \(\mathbb{X}\)) and \(f\colon \mathbb{X}\to \mathbb{R}\) continuous, then \[ V(n):=f(S^nx) ,\quad n\in\mathbb{Z} \] for some \(x\in\mathbb{X}\).
The key point of interest are now such operators which come from product dynamical systems; that is \(\mathbb{X} = \mathbb{X}_1 \times \mathbb{X}_2\) and \(S = S_1\times S_2\). Examples contained in this class may be sums of two potentials (e.g., periodic plus random, or periodic plus periodic with incommenuate frequencies) which are of the form \[ V(n) = V_1(n) + V_2(n) , \quad n\in\mathbb{Z}, \] where \(f(x^1,x^2):=f_1(x^1) + f_2(x^2)\) and \(f_j\) generates \(V_j\) as above, but also \(V\) generated by \[ f(x^1,x^2):= f_1(x^1)\cdot f_2(x^2), \] i.e., multiplicative modulation.
The paper investigates spectral results for such operator (or operator families) \(H\), in particular Cantor spectra of zero Lebesgue measure (sum of a potential generated by a locally constant function sampled over a minimal subshift satisfying Boshernitzans condition and a periodic potential; as well as their product instead of sum), absence of eigenvalues, representation of the spectrum by spectra of periodic modification of random potentials, and (sub-/super-)criticality in case of periodic perturbations of quasiperiodic potentials with applications of the almost Mathieu operator.