Scattering and Minimization Theory for Cubic Inhomogeneous Nls with Inverse Square Potential

In this paper, we study the scattering theory for the cubic inhomogeneous Schrodinger equations with inverse square potential iut + u - a |x|2 u =.|x|-b|u|2u with a > - 1 4 and 0 < b < 1 in dimension three. In the defocusing case (i.e.. = 1), we establish the global well-posedness and scattering for any initial data in the energy space H1 a (R3). While for the focusing case(i.e.. = -1), we 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 and Murphy (Proc Am Math Soc 145:4859-4867, 2017) and Campos and Cardoso (Proc Am Math Soc 150:2007-2021, 2022) that avoids the use of interaction Morawetz estimate. We also address the existence and the non-existence of normalized solutions of the above Schrodinger equation in dimension N for the focusing and defocusing cases.