The usual spectral decomposition of a Schrödinger operator roughly corresponds to a dynamical classification of the states into scattering states, bound states, and exotic states with less clear-cut dynamical properties. It used to be believed (according to what you can read in the books) that, therefore, only the first two types of states should occur for a physically reasonable system. It definitely came as a major surprise when David Pearson in 1978 presented a simple system with exotic states only (and, in fact, some structural stability). The picture has completely changed since, thanks to the work of Simon and others, who established the ubiquitous nature of these exotic states. Incidentally, the mechanism here is quite different from the one at work in the Pearson examples: One uses the fact that there are many operators with bound states only together with the extremely high structural instability of such operators.
If the system is in an exotic state, the particle spends most of the time far out. It may, however, return to its initial position infinitely often. Whether or not this actually happens is a question that apparently has to be analyzed case by case. From a mathematical point of view, this will usually be a rather subtle problem, with delicate dependence on small details. In more technical terms, one has to decide whether or not the spectral measure is a so-called Rajchman measure. This problem points out a particular model where the question looks both interesting and difficult.