Electron pair forming in an integrable model

We consider a lattice model in which phonons scatter with pairs of electrons. All eigenvalues and eigenvectors can be obtained analytically. For a suitable choice of parameters the ground state consists of a Fermi sea of non-interacting electrons, with on top a layer of paired electrons. The binding energy of the electron pairs is partly cancelled by an increased kinetic energy. This results in a gap in the kinetic energy spectrum of the electrons.