Cooper problem

Two added electrons move above a frozen filled Fermi sea and attract within a thin energy shell. The Pauli-blocked continuum has a logarithmic pair susceptibility, so an arbitrarily weak net attractive constant interaction produces a state below the two-particle threshold at zero temperature. This two-added-particle result is not by itself a full many-electron superconducting state.

For shell width $D$, attraction $g>0$ and constant single-spin density of states $\rho$, $1=(g\rho /2)\log [(2D+B)/B]$ gives $B=2D/[\exp (2/(g\rho ))-1]$. The pair energy relative to the free threshold is $-B$. At weak coupling this scale is nonanalytic in $g$.

The pairing Hamiltonian is projected onto empty states in a specified shell above the inert Fermi sea. A state restricted to that shell is an eigenstate of this projected model, not necessarily of a contact interaction that remains unrestricted at all momenta.