  We theoretically investigate the spectrum of elementary excitations of a
trapped dipolar quantum gas across the BEC-supersolid phase transition. Our
calculations reveal the existence of distinct Higgs and Nambu-Goldstone modes
that emerge from the softening roton modes of the dipolar BEC at the phase
transition point. On the supersolid side of the transition, the energy of the
Higgs mode increases rapidly, leading to a strong coupling to higher-lying
modes. Our study highlights how the symmetry-breaking nature of the supersolid
state translates to finite-size systems.
