Strichartz Estimates for Schr{ö}dinger equation with singular and time dependent Potential and Application to NLS

Abstract: We establish inhomogeneous Strichartz Estimates for the Schr{\"o}dinger equation with singular and time dependent potentials for non-admissible pairs. Our work extends the results provided by Vilela [23] and Foschi [6] where they proved the results in the absence of potential. It also extends the works of Pierfelice [20] and Burq, Planchon, Stalker, Tahvildar-Zadeh [3], who proved the estimates for admissible pairs. We also extend the recent work of Mizutani, Zhang, Zheng [17] and as an application of it, we improve the stability result of Kenig-Merle [13], which in turn establishes a proof (alternative to [26]) of existence of scattering solution for the energy critical focusing NLS with inverse square potential.