We discuss interacting and non-interacting one dimensional atomic systems
trapped in an optical lattice plus a parabolic potential. We show that, in the
tight-binding approximation, the non-interacting problem is exactly solvable in
terms of Mathieu functions. We use the analytic solutions to study the
collective oscillations of ideal bosonic and fermionic ensembles induced by
small displacements of the parabolic potential. We treat the interacting boson
problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From
analysis of the dependence upon lattice depth of the low-energy excitation
spectrum of the interacting system, we consider the problems of
"fermionization" of a Bose gas, and the superfluid-Mott insulator transition.
The spectrum of the noninteracting system turns out to provide a useful guide
to understanding the collective oscillations of the interacting system,
throughout a large and experimentally relevant parameter regime.
