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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 $L^2$-critical or $L^2$-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.