Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.10270 · Journal of Mathematical Physics

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati, Giulio Pasqualetti, Benjamin Schlein

Published Apr 1, 2022• 11 citations
arXivHTMLPDF

Abstract

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

DOI
10.1063/5.0087116

Related papers

A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati +2 more

Abstract We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius $$\mathfrak {a}/N$$ a / N , moving in the three-dimensional unit torus $$\Lambda $$ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $$N \rightarrow \infty $$ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.

arXiv:2203.11917•Communications in Mathematical Physics•Dec 5, 2022•18 citations

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

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.

Archive for Rational Mechanics and Analysis•Oct 21, 2024•9 citations

Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional

Élliott H. Lieb, Robert Seiringer, Jakob Yngvason

The ground-state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here a rigorous proof of the asymptotic exactness of this approximation for the ground-state energy and particle density of a dilute Bose gas with a positive interaction.

arXiv:math-ph/9908027•Physical Review A•Mar 6, 2000•375 citations

Bose gases in the Gross–Pitaevskii limit: A survey of some rigorous results

Benjamin Schlein

We review some mathematical work on the Bose gas in the Gross–Pitaevskii regime. We start with the classical results by Lieb, Seiringer and Yngvason on the ground-state energy and by Lieb and Seiringer on the existence of Bose–Einstein condensation. Afterwards, we discuss some more recent progress, based on a rigorous version of Bogoliubov theory.

arXiv:2203.10855•EMS Press eBooks•Jun 29, 2022•9 citations

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.

arXiv:2212.04431•Dec 8, 2022•3 citations

A second order upper bound on the ground state energy of a Bose gas beyond the Gross–Pitaevskii regime

Giulia Basti

We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.

arXiv:2203.01841•Journal of Mathematical Physics•Jul 1, 2022•4 citations
We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.
Abstract We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/N\mathfrak {a}/Na/N a / N , moving in the three-dimensional unit torus Λ\Lambda Λ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.
Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius a\mathfrak {a}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 4πρa4\pi \rho \mathfrak {a}4πρa 4 π ρ a and shows that corrections are smaller than Cρa(ρa3)1/2C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}Cρa(ρa3)1/2 C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant C>0C > 0C>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 ρa(ρa3)1/2\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}ρa(ρa3)1/2 ρ a ( ρ 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 a\frak{a}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 4πρa4πρ\frak{a}4πρa and shows that corrections are smaller than Cρa(ρa3)1/2C ρ\frak{a} (ρ\frak{a}^3)^{1/2}Cρa(ρa3)1/2, for a sufficiently large constant C>0C > 0C>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 ρa(ρa3)1/2ρ\frak{a} (ρ\frak{a}^3)^{1/2}ρa(ρa3)1/2, in agreement with the Lee-Huang-Yang prediction.
We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.