  We study the non-equilibrium dynamics of a one-dimensional Bose gas trapped
by a harmonic potential for a quench from zero to infinite interaction. The
different thermodynamic limits required for the equilibrium pre- and
post-quench Hamiltonians are the origin of a few unexpected phenomena that have
no counterparts in the translational invariant setting. We find that the
dynamics is perfectly periodic with breathing time related to the strength of
the trapping potential. For very short times, we observe a sudden expansion
leading to an extreme dilution of the gas and to the emergence of slowly
decaying tails in the density profile. The haste of the expansion induces a
undertow effect with a pronounced local minimum of the density at the center of
the trap. At half period there is a refocusing phenomenon characterized by a
sharp central peak of the density, juxtaposed to algebraically decaying tails.
We finally show that the time-averaged density is correctly captured by a
generalized Gibbs ensemble built with the conserved mode occupations.
