Dynamics of Threshold Solutions for Energy Critical NLS with Inverse Square Potential

We consider the focusing energy critical NLS with inverse square potential in d= 3, 4, 5 with the details given in d=3 and remarks on results other dimensions. Solutions on the energy surface of the ground state are. We prove that solutions with kinetic energy less than that of ground state must scatter to zero or belong to the stable/unstable of the ground state. In the latter case they converge to the ground exponentially in the energy space as t\ \ or t\ -\. (In3-dim without radial assumption, this holds under the compactness assumption of-scattering solutions on the energy surface. ) When the kinetic energy is than that of the ground state, we show that all radial H¹ solutions up in finite time, with the only two exceptions in the case of 5-dim which to the stable/unstable manifold of the ground state. The proof relies on detailed spectral analysis, local invariant manifold theory, and a global analysis.