  We test the time evolution of quite general initial states in a model that is
exactly solvable, $i.e.$ a semi-infinite $XY$ spin chain with an impurity at
the boundary. The dynamics is portrayed through the observation of the site
magnetization along the chain, focusing on the long-time behavior of the
magnetization, which is estimated using the stationary phase method. Localized
states are split off from the continuum for some regions of the impurity
parameter space. Bound states are essential for the non-ergodic behavior
reported here. When two impurity states exist, the quantum interference between
them leads to magnetization oscillations which settle over very long times with
the absence of damping. The frequency of the remanent oscillation is recognized
as being the Rabi frequency of the localized levels.
