The Petite Conjecture on almost periodicity and weakly localized solutions

A localized solution is a spatially localized solution, and regularity in space refers to the spatial regularity assumed in the surrounding discussion. A weakly localized solution (WLS) is a solution component that is not fully localized in the usual sense. The Petite Conjecture. A localized solution that is also regular in space is an almost periodic function of time. The source notes that weakly localized and self-similar solutions do not satisfy this property, and suggests that time-dependent potentials with chaotic time behavior, localized in space, may generically destroy bound states; the conjectural mechanism is associated with concentration of low-frequency modes near zero.