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arXiv:2311.07433 · Communications in Mathematical Physics

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

Cristina Caraci, Alessandro Olgiati, Diane Saint Aubin, Benjamin Schlein

Published Jun 3, 2025• 7 citations
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Abstract

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 .

DOI
10.1007/s00220-025-05320-7

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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.
The standard calculations of the ground-state energy of a homogeneous Bose gas rely on approximations which are physically reasonable but difficult to control. Lieb and Yngvason ( Phys. Rev. Lett. 80 (1998) 2504) have proved rigorously that the commonly accepted leading-order term of the ground-state energy is correct in the zero-density limit. Here, strong indications are given that also the next-to-leading term is correct. It is shown that the first terms obtained in a perturbative treatment provide contributions which are lost in the Bogoliubov approach.
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.