A Lower Bound on the Ground State Energy of Dilute Bose Gas

TL;DR

This paper establishes a new lower bound on the ground state energy per particle for a dilute three-dimensional Bose gas, refining understanding of its quantum behavior at low densities.

Contribution

It provides a rigorous lower bound for the ground state energy of dilute Bose gases, improving previous estimates by including correction terms involving the density.

Findings

Derived a lower bound matching the expected leading order with correction terms

Confirmed the asymptotic behavior of the ground state energy at low densities

Enhanced theoretical understanding of dilute Bose gas energetics

Abstract

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box $\Lambda$ of side length $L$. We take the limit $N, L \to \infty$ while keeping the density $\rho = N / L^3$ fixed and small. We prove a new lower bound for its ground state energy per particle $$\frac{E(N, \Lambda)}{N} \geq 4 \pi a \rho [ 1 - O(\rho^{1/3} |\log \rho|^3) ],$$ as $\rho \to 0$, where $a$ is the scattering length of $V$.