Exact BCS solution in the BCS-BEC crossover

The exact Richardson solution of the reduced BCS Hamiltonian is used to study the BCSto-BEC crossover, as well as the nature of Cooper pairs, in superconducting and Fermi superfluid media. Based on the exact eigenstate we will discuss the Cooper-pair concept proposing a scenario for the BCS-to-BEC crossover in which a mixture of quasifree fermions and pair resonances (BCS) evolves to a system of weakly bound molecules (BEC). In this single unified scenario the Cooper-pair wavefunction has a unique functional form. We propose a new definition of the condensate fraction which, within the limits of the BCS model, gives a qualitative description of recent experiments in ultracold atomic Fermi gases. Finally, we will introduce a new integrable model for asymmetric superfluid systems able to describe different homogeneous and inhomogeneous competing phases such as, breached superconductivity, deformed Fermi superfluidity, and the elusive Larkin- Ovchinnikov-Fulde-Ferrell (LOFF) state.