Ground State Energy of the Two-Component Charged Bose Gas

TL;DR

This paper investigates the precise ground state energy of the two-component charged Bose gas, confirming Dyson's conjecture that it is determined by a mean-field minimization problem, extending previous results from the one-component case.

Contribution

We verify Dyson's conjecture for the two-component charged Bose gas, establishing the exact constant in the ground state energy asymptotics using a mean-field approach.

Findings

Confirmed Dyson's conjecture for the two-component case

Established the exact constant in the energy asymptotics

Extended previous one-component results to a more complex two-component system

Abstract

We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for $N$ particles is at least as negative as $-CN^{7/5}$ for large $N$ and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant $C$ was given by a mean-field minimization problem that used, as input, Foldy's calculation (using Bogolubov's 1947 formalism) for the one-component gas. Earlier we showed that Foldy's calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson's conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.