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

In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential i u t + Δ u - a | x | 2 u = λ | x | - b | u | 2 u with a > - 1 4 and 0 < b < 1 in dimension three. In the defocusing case (i.e. λ = 1 ), we establish the global well-posedness and scattering for any initial data in the energy space H a 1 (R 3) . While for the focusing case(i.e. λ = - 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 Schrödinger equation in dimension N for the focusing and defocusing cases. [ABSTRACT FROM AUTHOR]