  Using Fourier transform on a time series generated by unitary evolution, we
extract many-body eigenstates of the Bose-Hubbard model corresponding to low
energy excitations, which are generated when the insulator-superfluid phase
transition is realized in a typical experiment. The analysis is conducted in a
symmetric external potential both without and with and disorder. A simple
classification of excitations in the absence disorder is provided. The
evolution is performed assuming the presence of the parity symmetry in the
system rendering many-body quantum states either symmetric or antisymmetric.
Using symmetry-breaking technique, those states are decomposed into elementary
one-particle processes.
