Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies

We consider solutions of the focusing cubic and quintic Gross–Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the L² -critical or L² -supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.