Upper bounds on the ground state energy of dilute hard core bosons

In the last decades, since the first experimental realizations of Bose- Einstein condensates, the study of large bosonic systems has been a very active field of research both in physics and in mathematics. In this talk we address a question strictly related to condensation, namely the characterization of the ground state energy. More precisely, we review some recent estimates on the ground state energy of a dilute gas of confined hard core bosons. We present a new upper bound for hard spheres in the thermodynamic limit where the number of particles and the size of the box are sent to infinity keeping the density fixed. Our result resolves the ground state energy up to an error of the order of the so-called Lee-Huang-Yang correction. We also discuss the Gross-Pitaevskii regime, in which N hard spheres with radius of order 1/N move on the unit torus; in this setting, we show an upper bound for the energy, capturing the correct second order correction.
Based on joint works with S. Cenatiempo, A. Giuliani, A. Olgiati, G. Pasqualetti, B. Schlein.