  We investigate the problem of two atoms interacting via a short range s-wave
potential in the presence of a deep optical lattice of arbitrary dimension $D$.
Using a tight binding approach, we derive analytical results for the properties
of the bound state and the scattering amplitude. We show that the tunneling
through the barriers induces a dimensional crossover from a confined regime at
high energy to an anisotropic three dimensional regime at low energy. The
critical value of the scattering length needed to form a two-body bound state
shows a logaritmic dependence on the tunneling rate for D=1 and a power law for
$D>1$. For the special case D=1, we also compare our analytical predictions
with exact numerics, finding remarkably good agreement.
