  Strongly-interacting ultra-cold atoms in tight-binding optical lattice
potentials provide an ideal platform to realize the fundamental Hubbard model.
Here, after outlining the elementary single particle solution, we review and
expand our recent work on complete characterization of the bound and scattering
states of two and three bosonic atoms in a one-dimensional optical lattice. In
the case of two atoms, there is a family of interaction-bound "dimer" states of
co-localized particles that exists invariantly for either attractive or
repulsive on-site interaction, with the energy below or above the two-particle
scattering continuum, respectively. Adding then the third particle -- "monomer"
-- we find that, apart from the simple strongly-bound "trimer" corresponding to
all three particles occupying the same lattice site, there are two peculiar
families of weakly-bound trimers with energies below and above the
monomer-dimer scattering continuum, the corresponding binding mechanism being
an effective particle exchange interaction.
