In systems of interacting fermions at low density, the ground state energy can be described by an expansion in terms of the density itself. The first order contribution does not depend on the interaction, while the second order contribution is proportional to the scattering length of the pair potential. The appearance of the scattering length is due to correlations in the system. For analogous systems of interacting bosons, these correlations have been described, up to the relevant order, by the action of a Bogoliubov transformation. Since couples of fermions behave almost like a boson, it is possible to write a fermionic Bogoliubov transformation, involving two fermionic excitations in place of each single bosonic excitation. This idea provides a new strategy to obtain thecoefficient for the second order contribution to the ground state energy in a rigorous fashion, and with improved bounds on the remainder. Based on a joint work with E.L. Giacomelli, C. Hainzl, and M. Porta.
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