I will discuss the Bogolubov variational model that arises when a many-body bosonic Hamiltionian is restricted to quasi-free states. For simplicity I will restrict attention to ground states. By definition the variational model gives an upper bound on the many-body ground state energy. I will discuss examples where it is, in fact, a good approximation, such as the Lieb-Liniger model, the one- and two-component charged Bose gases and the dilute Bose gas in three dimensions.
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