We establish localization type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with one-frequency quasiperiodic potentials defined by piecewise Hölder functions. This, in particular, extends some results previously known only for trigonometric polynomials [9] to the case of surprisingly low regularity. On the technical level, an important part of the argument is an extension of uniform uppersemicontinuity to cocycles with discontinuities, a result of independent interest.

We study discrete quasiperiodic Schrödinger operators on ℓ 2 (Z) with potentials defined by γ-Hölder functions. We prove a general statement that for γ > 1/2 and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.

The Holstein model describes the motion of a tight-binding tracer particle interacting with a field of quantum harmonic oscillators. We consider this model with an on-site random potential. Provided the hopping amplitude for the particle is small, we prove localization for matrix elements of the resolvent, in particle position and in the field Fock space. These bounds imply a form of dynamical localization for the particle position that leaves open the possibility of resonant tunneling in Fock space between equivalent field configurations.

We prove localization and probabilistic bounds on the minimum level spacing for a random block Anderson model without monotonicity. Using a sequence of narrowing energy windows and associated Schur complements, we obtain detailed probabilistic information about the microscopic structure of energy levels of the Hamiltonian, as well as the support and decay of eigenfunctions.Comment: 39 pages, 3 figures, minor updates for published version. To appear in Jour. Stat. Phy

We consider resonant tunneling between disorder localized states in a potential energy displaying perfect correlations over large distances. The phenomenon described here may be of relevance to models exhibiting many-body localization. Furthermore, in the context of single particle operators, our examples demonstrate that exponential resolvent localization does not imply exponential dynamical localization for random Schrödinger operators with correlated potentials.