  We consider quantum quenches of harmonically trapped one-dimensional bosons
from repulsive to attractive interactions, and the resulting breathing dynamics
of the so-called `super-Tonks-Girardeau' (sTG) state. This state is highly
excited compared to the ground state of the attractive gas, and is the lowest
eigenstate where the particles are not bound or clustered. We analyze the
dynamics from a spectral point of view, identifying the relevant eigenstates of
the interacting trapped many-body system, and analyzing the nature of these
quantum eigenstates. To obtain explicit eigenspectra, we use Hamiltonians with
finite-dimensional Hilbert spaces to approximate the Lieb-Liniger system. We
employ two very different approximate approaches: an expansion in a truncated
single-particle harmonic-trap basis and a lattice (Bose-Hubbard) model. We show
how the breathing frequency, identified with the energy difference between the
sTG state and another particular eigenstate, varies with interaction.
