Quantum revivals and fractality for the Schr\"odinger equation

We investigate the behavior of the Schr\"odinger equation under the influence of potentials, focusing on its relationship to quantum revivals and fractality. Our findings reveal that the solution displays fractal behavior at irrational times, while exhibiting regularity similar to the initial data at rational times. These extend the results of Oskolkov \cite{O} and Rodnianski \cite{R2} on the free Schr\"odinger evolution to the general case regarding potentials.Comment: 14page

Sharp Lp−LqL^{p}-L^{q} estimates for the spherical harmonic projection (Harmonic Analysis and Nonlinear Partial Differential Equations)

"Harmonic Analysis and Nonlinear Partial Differential Equations". July 3～5, 2017. edited by Hideo Takaoka and Hideo Kubo. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider Lp-Lq estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of p, q. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which extends the earlier results due to Jerison and Kenig [22], and Stein [34]