Global-in-time smoothing effects for Schr\"odinger equations with inverse-square potentials

The purpose of this note is to prove global-in-time smoothing effects for the Schr"odinger equation with potentials exhibiting critical singularity. A typical example of admissible potentials is the inverse-square potential $a\mathrm{\mid }x{\mathrm{\mid }}^{-2}$ with $a>-(n-2{)}^2\mathrm{/}4$. This particularly gives an affirmative answer to a question raised by Bui-D’Ancona-Duong-Li-Ly [4]. The proof employs a uniform resolvent estimate proved by Barcel'o-Vega-Zubeldia [1] and an abstract perturbation method by Bouclet-Mizutani [3].