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Ground state energy of the low density Bose gas with two-body and three-body interactions

François L. A. Visconti

TL;DR

This work proves that the leading ground-state energy per volume of a dilute Bose gas with both two-body and three-body interactions is governed by the two-body scattering length $a(V)$ and the three-body scattering energy $b_{ ilde{ ext{M}}}(W)$, yielding $e( ho,V,W)=igl(4 ext{π} a(V) ho^2+ frac{1}{6} b_{ ilde{ ext{M}}}(W) ho^3igr)(1+O(Y^ u))$ in the dilute limit with $Y= ho ilde{ ext{a}}^3$ and $ ilde{ ext{a}}= ext{max}(a(V), ho b_{ ilde{ ext{M}}}(W))$. The proof combines a robust lower-bound strategy based on a multi-scale Dyson lemma and Temple’s inequality with a matching upper-bound construction using a Jastrow-type trial state incorporating both two- and three-body correlations; the analysis carefully decouples the two-body and three-body contributions despite their coexistence. This work extends classical two-body results (Dyson, Lieb–Yngvason) and the recent three-body theory (Nam–Ricaud–Triay), resolving a conjecture for the mixed interaction case. The results clarify the role of distinct length scales (two-body and three-body Gross–Pitaevskii scales) and provide a rigorous foundation for the leading-order energy in experiments where both interaction types are relevant. Implications include improved understanding of dilute Bose condensates stabilized by three-body effects and guidance for mean-field-type limits (e.g., GP and beyond) with composite interactions.

Abstract

In the present paper we study the low density Bose gas in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the ground state energy is entirely characterised by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) and of Lieb-Yngvason (1998), and the result for three-body interactions was proven by Nam-Ricaud-Triay (2022). The present result resolves a conjecture of Nam-Ricaud-Triay (2022).

Ground state energy of the low density Bose gas with two-body and three-body interactions

TL;DR

This work proves that the leading ground-state energy per volume of a dilute Bose gas with both two-body and three-body interactions is governed by the two-body scattering length and the three-body scattering energy , yielding in the dilute limit with and . The proof combines a robust lower-bound strategy based on a multi-scale Dyson lemma and Temple’s inequality with a matching upper-bound construction using a Jastrow-type trial state incorporating both two- and three-body correlations; the analysis carefully decouples the two-body and three-body contributions despite their coexistence. This work extends classical two-body results (Dyson, Lieb–Yngvason) and the recent three-body theory (Nam–Ricaud–Triay), resolving a conjecture for the mixed interaction case. The results clarify the role of distinct length scales (two-body and three-body Gross–Pitaevskii scales) and provide a rigorous foundation for the leading-order energy in experiments where both interaction types are relevant. Implications include improved understanding of dilute Bose condensates stabilized by three-body effects and guidance for mean-field-type limits (e.g., GP and beyond) with composite interactions.

Abstract

In the present paper we study the low density Bose gas in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the ground state energy is entirely characterised by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) and of Lieb-Yngvason (1998), and the result for three-body interactions was proven by Nam-Ricaud-Triay (2022). The present result resolves a conjecture of Nam-Ricaud-Triay (2022).
Paper Structure (13 sections, 8 theorems, 147 equations, 1 figure)

This paper contains 13 sections, 8 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model presentation
  3. Dilute regime
  4. Main result
  5. Strategy of the proof
  6. Scattering energy
  7. Truncated two-body scattering solution
  8. Three-body scattering problem
  9. Lower bound
  10. Dyson Lemmas
  11. Proof of Proposition \ref{['prop:energy_short_length']}: Energy at short length scales
  12. Proof of the lower bound in Theorem \ref{['th:main_result']}: division into small boxes
  13. Upper bound

Key Result

Theorem 2

Let $V$ and $W$ satisfy Assumption assumption:potentials. Define $Y \coloneqq \rho\mathfrak{a}^3$ with $\mathfrak{a}\coloneqq \max(a(V),\rho b_\mathcal{M}(W))$. Then, under the assumption that $R_0\mathfrak{a}^{-1}$ is bounded, the thermodynamic ground state energy per unit volume defined in eq:ther in the dilute limit $Y\rightarrow 0$, for some universal constant $\nu>0$ (independent of $V$ and $

Figures (1)

  • Figure 1: Illustration of the two possible configurations for which the l.h.s of \ref{['eq:no_three_and_four_body_collision_bound']} is non zero. Both configurations are incompatible.

Theorems & Definitions (15)

  • Theorem 2
  • Lemma 3: Truncated two-body scattering solution
  • Lemma 4: Truncated three-body scattering solution
  • proof
  • Proposition 5: Energy at short length scales
  • Lemma 6: Dyson Lemma for radial potentials
  • proof
  • Lemma 7: Dyson lemma for three-body potentials
  • proof
  • Lemma 8: Many-body Dyson lemma with two-body and three-body interactions
  • ...and 5 more