We consider the Cauchy problem for the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of the kinetic energy and potential energy, is conserved for all time. Due to the focusing nature, solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or even have mixed behaviors. Our goal in this thesis is to fully characterize the solution when the energy is below or at the level of the energy of the ground state solution $W_a$. Our main result contains two parts.
First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters.
Second, we show a rigidity result at the level of ground state solution. We prove that among all solutions with the same energy as the ground state solution, there are only two (up to symmetries) solutions $W_a^+, W_a^-$ that are exponential close to $W_a$ and serve as the threshold of scattering and blow-up. All solutions with the same energy will blow up both forward and backward in time if they go beyond the upper threshold $W_a^+$; all solutions with the same energy will scatter both forward and backward in time if they fall below the lower threshold $W_a^-$.
In the case of NLS with no potential, this type of results was first obtained by Kenig-Merle \cite{R: Kenig focusing} and Duyckaerts-Merle \cite{R: D Merle}. However, as the potential has the same scaling as $\Delta$, one can not expect to extend their results in a simple perturbative way. We develop crucial spectral estimates for the operator $-\Delta+a/|x|^2$, we also rely heavily on the recent understanding of the operator $-\Delta+a/|x|^2$ in \cite{R: Harmonic inverse KMVZZ}.