The energy of dilute Bose gases II: The general case

TL;DR

This paper rigorously establishes a lower bound for the ground state energy density of dilute Bose gases that aligns with the Lee-Huang-Yang formula, including cases with large interaction potentials like hard cores.

Contribution

It provides a mathematically rigorous proof of the lower bound for the energy density of dilute Bose gases, covering cases with large potentials, solving a longstanding problem in mathematical physics.

Findings

Lower bound matches the Lee-Huang-Yang formula.

Includes potentials with large L^1-norm, such as hard core interactions.

Addresses a major open problem since the 1960s.

Abstract

For a dilute system of non-relativistic bosons interacting through a positive potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(\rho) \geq 4\pi a \rho^2 (1+ \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} +o(\sqrt{\rho a^3}\,))$, thereby proving a lower bound consistent with the Lee-Huang-Yang formula for the energy density. The proof allows for potentials with large $L^1$-norm, in particular, the case of hard core interactions is included. Thereby, we solve a problem in mathematical physics that had been a major challenge since the 1960's.