Normalized ground states and a mass-constrained scattering threshold for the inhomogeneous NLS with an inverse-square potential

We study the focusing inhomogeneous nonlinear Schrödinger equation with an inverse-square potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-a\mathrm{\mid }x{\mathrm{\mid }}^{-2}u+\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$ in the mass-supercritical and energy-subcritical regime $$\frac{4-2b}{N}<\alpha <\frac{4-2b}{N-2},$$ where $N\ge 3$ and $0<b<2$. Two main lines of research have developed largely independently for this model: scattering theory below the ground-state threshold and the variational analysis of solutions with prescribed mass. In this work, we establish a connection between these two approaches.