Bose–Einstein condensation at fixed density
For a homogeneous three-dimensional Bose gas with a fixed nonnegative finite-range two-body potential, prove at zero temperature that the largest eigenvalue of the one-body density matrix is at least in the thermodynamic limit , for sufficient
Problem statement
For a homogeneous three-dimensional Bose gas with a fixed nonnegative finite-range two-body potential, prove at zero temperature that the largest eigenvalue of the one-body density matrix is at least in the thermodynamic limit , for sufficiently small fixed gas parameter and some .
Lean 4
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Background
Use periodic boxes, fixed kinetic-energy normalization and a potential with positive scattering length . The density and potential stay fixed as the box grows: this is not a mean-field or Gross–Pitaevskii scaling limit.
Established baseline: rigorous energy asymptotics and condensation theorems in other scaling regimes provide substantial progress but do not imply a macroscopic one-body eigenvalue in this homogeneous fixed-density limit. Solovej's 2025 account explicitly separates the energy expansion from this condensation question.
Positive-temperature condensation and quantitative depletion are further targets, not required for the zero-temperature existence benchmark.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Solovej, original open questions on the dilute Bose gas (2025), sections 5 and 8 link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
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Discussion
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