  We investigate the breathing mode of harmonically trapped bosons in an
optical lattice at small site occupancies. The Bose-Hubbard model with a
trapping potential is used to describe the breathing-mode dynamics initiated
through weak quenches of the trap strength. We connect to results for continuum
bosons (Lieb-Liniger and Gross-Pitaevskii results) and also present deviations
from continuum physics. We take a spectral perspective, identifying the
breathing mode frequency with a particular energy gap in the spectrum of the
trapped Bose-Hubbard Hamiltonian. We present the low energy eigenspectrum of
the trapped many-boson system, and study overlaps of the initial state with
eigenstates of the quenched Hamiltonian. There is an intermediate interaction
regime, between a "free-boson" limit and a "free-fermion" limit, in which the
Bose-Hubbard breathing mode frequency approaches the Gross-Pitaevskii
prediction. In addition, we present a striking failure of the time-dependent
Gutzwiller approximation for describing breathing modes.
