  Probing the radial collective oscillation of a trapped quantum system is an
accurate experimental tool to investigate interactions and dimensionality
effects. We consider a fully polarized quasi-one dimensional dipolar quantum
gas of bosonic dysprosium atoms in a parabolic trap at zero temperature. We
model the dipolar gas with an effective quasi-one dimensional Hamiltonian in
the single-mode approximation, and derive the equation of state using a
variational approximation based on the Lieb-Liniger gas Bethe Ansatz
wavefunction or perturbation theory. We calculate the breathing mode
frequencies while varying polarization angles by a sum-rule approach, and find
them in good agreement with recent experimental findings.
