Third order upper bound for the ground state energy of the dilute Bose gas

We derive an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit which correctly resolves the third order term predicted by Wu, Hugenholtz and Pines, and Sawada, with an error of the correct fourth order. Our result applies to compactly supported and radial potential that have positive scattering length, are stable, and do not admit two-body bound states. In particular, this includes the case of a positive potential, and thus generalises the technical assumptions of the Lee-Huang-Yang upper bound established by Yau and Yin, as well as the one by Basti, Cenatiempo, and Schlein. The most significant novelty in our proof is the introduction of a cutoff that constrains the local number of excitations and permits the construction of a trial state with correct energy density at any length scale below the thermodynamic one.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.