  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.
