Journal of Mathematical Physics

A note on the Wu’s term in the ground state energy of a Bose gas in the Gross–Pitaevskii regime

Cristina Caraci

Published Mar 1, 2026• 0 citations

Abstract

We consider a gas of N interacting bosons in the three dimensional unit box in the Gross-Pitaevskii regime. We review the derivation of the third order correction for the ground state energy, of order log N/N, as obtained in Caraci et al., [Commun. Math. Phys. 406, 153 (2025)], explaining the key steps of the proof. In addition, we show how to obtain the optimal error term of order O(1/N).

DOI: 10.1063/5.0310157

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Journal of Mathematical Physics

A note on the Wu’s term in the ground state energy of a Bose gas in the Gross–Pitaevskii regime

Cristina Caraci

Published Mar 1, 2026• 0 citations

Publisher

Abstract

DOI: 10.1063/5.0310157

Related papers

Third Order Corrections to the Ground State Energy of a Bose Gas in the Gross–Pitaevskii Regime

Cristina Caraci, Alessandro Olgiati, Diane Saint Aubin +1 more

Abstract For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy $E_N$ E N . While the leading, order N , contribution to $E_N$ E N has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of $E_N$ E N , which is of the order $(\log N) / N$ ( log N ) / N .Abstract For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy $$E_N$$ E N . While the leading, order N , contribution to $$E_N$$ E N has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of $$E_N$$ E N , which is of the order $$(\log N) / N$$ ( log N ) / N .

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

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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.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.

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We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross–Pitaevskii length scale. The presentation includes both the two- and three-dimensional cases, and catches the second-order correction, i.e. the Lee–Huang–Yang term. The calculation on a box of this length scale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross–Pitaevskii length scale. The presentation includes both the two- and three-dimensional cases, and catches the second-order correction, i.e. the Lee–Huang–Yang term. The calculation on a box of this length scale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.

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Abstract We provide a second-order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud, and Triay in [25], where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order $\sqrt {N}$ . In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order $N^{-\frac {3}{4}}$ .Abstract We provide a second-order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud, and Triay in [25], where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order $\sqrt {N}$ . In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order $N^{-\frac {3}{4}}$ .

arXiv:2406.19886Forum of Mathematics SigmaJan 1, 20250 citations

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.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:2203.11917Communications in Mathematical PhysicsDec 5, 202218 citations

Abstract For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy

E_N

E N . While the leading, order N , contribution to

E_N

E N has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of

E_N

E N , which is of the order

(\log N) / N

( log N ) / N .

\sqrt {N}

. In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order

N^{-\frac {3}{4}}

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.