Off-diagonal long-range order in a many-body model of attractive fermions
In plain words
BCS theory explains superconductivity by electron pairing, but it is an approximation. No one has proven that a full many-electron model with attraction actually forms a pair condensate in an infinite system.
Precise statement
$N$ spin-$1/2$ fermions in a box of side $L$ with attractive short-range pair potential, density fixed, $L \to \infty$. Prove that the two-body reduced density matrix has an eigenvalue of order $N$ (Yang's off-diagonal long-range order) for $T$ below a $T_{c} > 0$, and compare $T_{c}$ with the BCS functional prediction. Answer: a proof.
What would settle it
A proof of pair condensation in the thermodynamic limit for the full many-body Hamiltonian.
Status in the literature
Rigorous results concern the BCS functional (Hainzl, Hamza, Seiringer and Solovej, 2008, and later) and mean-field type Hamiltonians, not the full many-body problem.