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.
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