  We study the behavior of the excitation spectrum across the quantum phase
transition from a superfluid to a supersolid phase of a dipolar Bose gas
confined to a one-dimensional geometry. Including the leading beyond-mean-field
effects within an effective Hamiltonian, the analysis is based on Bogoliubov
theory with several order parameters accounting for the superfluid as well as
solid structure. We find fast convergence of the ground-state energy in the
supersolid with the number of order parameters and demonstrate a stable
excitation spectrum with two Goldstone modes and an amplitude mode in the
low-energy regime. Our results suggest that there exists an experimentally
achievable parameter regime for dysprosium atoms, where the supersolid phase
exhibits a stable excitation spectrum in the thermodynamic limit and the
transition into the supersolid phase is of second order driven by the roton
instability.
