The focusing cubic NLS with inverse-square potential in three space   dimensions
  We consider the focusing cubic nonlinear Schr\"odinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold. 
Rowan Killip, Jason Murphy, Monica Visan, Jiqiang Zheng
math.AP