  The lowest-lying collective modes of a trapped Bose gas in an optical lattice
are studied in the Bose-Hubbard model. An exact diagonalization of the
Hamiltonian is performed in a one-dimensional five-particle system in order to
find the lowest few eigenstates. Dipole and breathing character of the
eigenstates is confirmed in the limit where the tunneling dominates the
dynamics, but under Mott-like conditions the excitations do not correspond to
oscillatory modes.
