arXiv:2101.06222 · Forum of Mathematics Sigma

A new second-order upper bound for the ground state energy of dilute Bose gases

Giulia Basti, Serena Cenatiempo, Benjamin Schlein

Published Jan 1, 2021• 44 citations

Abstract

Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb {R}^3)$ , gives better rates and, in our opinion, is substantially simpler.

DOI: 10.1017/fms.2021.66

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arXiv:2101.06222 · Forum of Mathematics Sigma

A new second-order upper bound for the ground state energy of dilute Bose gases

Giulia Basti, Serena Cenatiempo, Benjamin Schlein

Published Jan 1, 2021• 44 citations

arXiv HTML PDF

Abstract

V \in L^3 (\mathbb {R}^3)

, gives better rates and, in our opinion, is substantially simpler.

DOI: 10.1017/fms.2021.66

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arXiv:2304.02405Apr 5, 20233 citations

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Upper bound for the ground state energy of a dilute Bose gas of hard spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4πρ\frak{a}$ and shows that corrections are smaller than $C ρ\frak{a} (ρ\frak{a}^3)^{1/2}$ , for a sufficiently large constant $C > 0$ . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $ρ\frak{a} (ρ\frak{a}^3)^{1/2}$ , in agreement with the Lee-Huang-Yang prediction.We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4πρ\frak{a}$ and shows that corrections are smaller than $C ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, for a sufficiently large constant $C > 0$. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, in agreement with the Lee-Huang-Yang prediction.

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V \in L^3 (\mathbb {R}^3)

, gives better rates and, in our opinion, is substantially simpler.

We consider a gas of bosons interacting through a hard-sphere potential with radius

\frak{a}

in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4πρ\frak{a}

and shows that corrections are smaller than

C ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, for a sufficiently large constant

C > 0

. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order

ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, in agreement with the Lee-Huang-Yang prediction.

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius

\mathfrak {a}

a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4\pi \rho \mathfrak {a}

4 π ρ a and shows that corrections are smaller than

C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}

C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant

C > 0

C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order

\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}

ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

We consider a quantum gas consisting of

N

hard spheres with radius

\frak{a} > 0

, obeying bosonic statistics and moving in the box

Λ= [0;L]^3

with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with

N, L \to \infty

at fixed density

ρ= N / L^3

. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit

ρ\frak{a}^3 \ll 1