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We explore the ground state properties of cold atomic gases, loaded into a bichromatic lattice, focusing on the cases of non-interacting fermions and hard-core (Tonks-Girardeau) bosons, trapped by the combination of two potentials with incommensurate periods. For such systems, two limiting cases have been thoroughly established. In the tight-binding limit, the single-particle states in the lowest occupied band show a localization transition, as the strength of the second potential is increased above a certain threshold. In the continuous limit, when the tight-binding approximation does not hold anymore, a mobility edge is found, whose position in energy depends upon the strength of the second potential. Here, we study how the crossover from the discrete to the continuum behavior occurs, and prove that signatures of the localization transition and mobility edge clearly appear in the generic many-body properties of the systems. Specifically, we evaluate the momentum distribution, which is a routinely measured quantity in experiments with cold atoms, and demonstrate that, even in the presence of strong boson-boson interactions, the single particle mobility edge can be observed in the ground state properties.

We investigate the dynamical properties of an interacting many-body system with a nontrivial energy potential landscape that may induce a singular continuous single-particle energy spectrum. Focusing on the Aubry-André model, whose anomalous transport properties in the presence of interaction was recently demonstrated experimentally in an ultracold-gas setup, we discuss the anomalous slowing down of the dynamics it exhibits and show that it emerges from the singular-continuous nature of the single-particle excitation spectrum. Our study demonstrates that singular-continuous spectra can be found in interacting systems, unlike previously conjectured by treating the interactions in the mean-field approximation. This, in turns, also highlights the importance of the many-body correlations in giving rise to anomalous dynamics, which, in many-body systems, can result from a nontrivial interplay between geometry and interactions.