Joint BCAM-UPV/EHU Analysis and PDE seminar: Scattering theory for cubic inhomogeneous NLS with inverse square potential

In this talk, we will discuss the scattering theory for the cubic inhomogeneous Schr\"odinger equations with inverse square potential iut+Δ u-a/|x|2 u=λ |x|-b|u|2u​​ with a>-1/4 ​​ and 0<b<1​​ in dimension three. In the defocusing case (i.e. λ=1​​), the global well-posedness and scattering for any initial data in the energy space H1a(ℝ3)​​ can be established. While for the focusing case(i.e. λ=-1​​), we can obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy and Campos-Cardoso that avoids the use of interaction Morawetz estimate.