  The collective field formulation of the Calogero model supports periodic
density waves. An important set of such density waves is a two-parameter family
of BPS solutions of the equations of motion of the collective field theory. One
of these parameters is essentially the average particle density, which
determines the period, while the other parameter determines the amplitude.
These BPS solutions are sometimes referred to as "small amplitude waves" since
they undulate around their mean density, but never vanish. We present complete
analysis of quadratic fluctuations around these BPS solutions. The
corresponding fluctuation hamiltonian (i.e., the stability operator) is
diagonalized in terms of bosonic creation and annihilation operators which
correspond to the complete orthogonal set of Bloch-Floquet eigenstates of a
related periodic Schr\"odinger hamiltonian, which we derive explicitly.
Remarkably, the fluctuation spectrum is independent of the parameter which
determines the density wave's amplitude. As a consequence, the sum over
zero-point energies of the field-theoretic fluctuation hamiltonian, and its
ensuing normal-ordering and regularization, are the same as in the case of
fluctuations around constant density background, namely, the ground state.
Thus, quadratic fluctuations do not shift the energy density tied with the
BPS-density waves studied here, compared to its ground state value. Finally, we
also make some brief remarks concerning fluctuations around non-BPS density
waves
