If a quantum mechanical system is in a scattering state, its state is a wave packet obtained by integrating over an energy interval. The building blocks are ideal states (which do not lie in the Hilbert space) with sharp energy. If nothing special happens, one expects that these ideal states are spread out uniformly over the whole space and oscillate in a regular way.

This is the last of three problems addressing these issues - compare The Schrödinger Conjecture and Asymptotic Phase Existence and Uniform Distribution. We are now interested in a combination of size and oscillation properties. More precisely, we ask whether the converse of a result of Pearson and Al-Naggar holds. These authors established existence of scattering states if there are (very) regularly oscillating ideal states and all states are of roughly the same size.