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.

Adami , G. Golse , A. Aizenman , E. Lieb , R. Seiringer , J. Solovej , J.

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Yngvason , Bose—Einstein quantum phase transition in an optical lattice model , Phys. A 70 , Zbl Anapolitanos , I. Bach , T. Chen , J. Sigal , Smooth Feshbach map and operator-theoretic renormalization group methods , J. Notes vol.

## On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations

Chen , N. A 27 no. Elgart , L. Schlein , H. Yau , Gross—Pitaevskii equation as the mean field limit of weakly coupled bosons , Arch. Elgart , B. Schlein , Mean field dynamics of boson stars , Comm. Pure Appl. Graffi , S.

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