Observability and unique continuation inequalities for the Schr\"{o}dinger equations with inverse-square potentials

In this paper, we focus on the Schr\"{o}dinger equations with inverse-square potentials in dimension one; these special potentials play an important role in the field of mathematical physics. We study several observability and unique continuation inequalities at one time point or at two time points for these equations. These observability and unique continuation inequalities are some new types of quantitative estimates which have appeared in recent literature. Their proofs essentially rely on the representation of the solution, a Nazarov-type uncertainty principle for the Hankel transform, and an interpolation inequality for functions whose Hankel transforms have compact support. Meanwhile, these inequalities can be applied to the controllability of these Schr\"{o}dinger equations.