Math Conjectures

Homogeneous interacting Bose-gas condensation

OPENMajorConjectureProposed 1925 · Standard version

Canonical statement

Let v≥0v\ge0 be a nonzero radial finite-range potential on R3\mathbb R^3 with scattering length aa, let ΛL=(R/LZ)3\Lambda_L=(\mathbb R/L\mathbb Z)^3, and let vLv_L be the LL-periodic extension of vv. For a normalized bosonic ground state of
HN,L=−∑i=1NΔi+∑i<jvL(xi−xj)on Lsym2(ΛLN), H_{N,L}=-\sum_{i=1}^N\Delta_i+\sum_{i<j}v_L(x_i-x_j) \quad\text{on }L^2_{\mathrm{sym}}(\Lambda_L^N),
let γN,L(1)\gamma^{(1)}_{N,L} be its one-particle density matrix, normalized by Tr⁡γN,L(1)=N\operatorname{Tr}\gamma^{(1)}_{N,L}=N. Then
lim⁡ρa3↓0 lim⁡N,L→∞N/L3=ρλmax⁡(γN,L(1))N=1. \lim_{\rho a^3\downarrow0}\ \lim_{\substack{N,L\to\infty\\N/L^3=\rho}}\frac{\lambda_{\max}(\gamma^{(1)}_{N,L})}{N}=1.
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Let \(v\ge0\) be a nonzero radial finite-range potential on \(\mathbb R^3\) with scattering length \(a\), let \(\Lambda_L=(\mathbb R/L\mathbb Z)^3\), and let \(v_L\) be the \(L\)-periodic extension of \(v\). For a normalized bosonic ground state of \[ H_{N,L}=-\sum_{i=1}^N\Delta_i+\sum_{i<j}v_L(x_i-x_j) \quad\text{on }L^2_{\mathrm{sym}}(\Lambda_L^N), \] let \(\gamma^{(1)}_{N,L}\) be its one-particle density matrix, normalized by \(\operatorname{Tr}\gamma^{(1)}_{N,L}=N\). Then \[ \lim_{\rho a^3\downarrow0}\ \lim_{\substack{N,L\to\infty\\N/L^3=\rho}}\frac{\lambda_{\max}(\gamma^{(1)}_{N,L})}{N}=1. \]
MATH-PHYS-010:[Einstein1925QuantentheorieGas][LiebSeiringer2002BECTrapped][LiebSeiringerSolovejYngvason2005BoseGas][BrenneckeBrooksCaraciOldenburg2025BEC]
Mathematical physicsAnalysis and functional analysisPartial differential equations

Notes

This problem asks for a proof of Bose–Einstein condensation for a genuinely interacting gas. For bosons on a torus with a nonzero, nonnegative, finite-range radial pair potential of scattering length aa, the conjecture asserts that in the thermodynamic limit at fixed density ρ\rho, followed by the dilute limit ρa3→0\rho a^3\to0, the largest eigenvalue of the ground state's one-particle density matrix, divided by NN, tends to 11: a single one-particle state is macroscopically occupied. The 1925 date refers to Einstein's prediction of condensation for the ideal gas [Einstein1925QuantentheorieGas]; the interacting dilute formulation developed from Bogoliubov's 1947 theory.

For the ideal gas condensation is classical, and the interacting problem has been settled in important but non-thermodynamic regimes: Lieb and Seiringer proved BEC for dilute trapped gases [LiebSeiringer2002BECTrapped], and the Gross–Pitaevskii scaling is by now well understood and has been pushed somewhat beyond [BrenneckeBrooksCaraciOldenburg2025BEC]. The monograph [LiebSeiringerSolovejYngvason2005BoseGas] surveys these results together with mean-field scalings and special gapped models where condensation is known.

What is missing is an off-diagonal-long-range-order estimate for a fixed interaction at fixed positive density in the translation-invariant thermodynamic limit — open even at zero temperature.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.