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Mathematical physics of dilute Bose gases

Jan Philip Solovej

TL;DR

The paper surveys rigorous mathematical results for dilute Bose gases across dimensions, focusing on the thermodynamic limit of the ground-state energy density $e(\rho)$ and its universal two-term expansions governed by the scattering length $a$. It develops a Bogoliubov-based variational framework that, together with careful handling of the cubic interaction terms, yields the leading term $4\pi\rho^2 a$ in 3D and the Lee-Huang-Yang correction, while analogous results in $d=2$ and $d=1$ are discussed with the appropriate asymptotics. The work also clarifies the status of Bose-Einstein condensation in both finite-size (Gross-Pitaevskii) and thermodynamic settings, and highlights key open problems, including the hard-core three-dimensional case, negative potentials, and complete thermodynamic BEC proofs. Overall, it demonstrates a universal dependence of the dilute energy on the scattering length and provides a rigorous bridge between many-body quantum mechanics and effective theories used in cold-atom physics.

Abstract

We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee-Huang-Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.

Mathematical physics of dilute Bose gases

TL;DR

The paper surveys rigorous mathematical results for dilute Bose gases across dimensions, focusing on the thermodynamic limit of the ground-state energy density and its universal two-term expansions governed by the scattering length . It develops a Bogoliubov-based variational framework that, together with careful handling of the cubic interaction terms, yields the leading term in 3D and the Lee-Huang-Yang correction, while analogous results in and are discussed with the appropriate asymptotics. The work also clarifies the status of Bose-Einstein condensation in both finite-size (Gross-Pitaevskii) and thermodynamic settings, and highlights key open problems, including the hard-core three-dimensional case, negative potentials, and complete thermodynamic BEC proofs. Overall, it demonstrates a universal dependence of the dilute energy on the scattering length and provides a rigorous bridge between many-body quantum mechanics and effective theories used in cold-atom physics.

Abstract

We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee-Huang-Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.

Paper Structure

This paper contains 8 sections, 1 theorem, 33 equations.

Table of Contents

  1. Introduction
  2. Scattering length
  3. Main result on energy density asymptotics
  4. Second quantization and Bogolyubov Theory
  5. Bose-Einstein Condensation
  6. Ideas behind the proof of the Lee-Huang-Yang formula
  7. Some reflections on the one-dimensional case
  8. Conclusion and summary of open problems

Key Result

theorem 1

For $v\geq 0$ with finite range in the ball $|x|\leq R$ we find in the limit $\rho|a|^d\to 0$ that The result in dimension three requires $v\in L^3$, i.e., that the $|v|^3$ is integrable, but this is only required to prove the upper bound in the estimate. Above $\Gamma$ is the Euler–Mascheroni constant.

Theorems & Definitions (1)

  • theorem 1: Ground state energy density asymptotics