Recently much interest has been given to the study of dilute quantum gases and their ground state energies. I will present recent work on two such problems: that of a spin-polarized and that of a spin-1/2 Fermi gas. In both settings we find (as an upper bound) the leading correction to the kinetic energy. This correction depends on the interaction only through its p- or s-wave scattering length: the p-wave in the spin-polarized case and the s-wave in the spin-1/2 case. One of the main ingredients in the proofs is a rigorous version of a formal cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237-260). I will discuss this expansion and the analysis of its absolute convergence.
Joint work with Robert Seiringer. Based on arXiv:2301.04894 and 2301.08005.