Lorentzian, or Lorentz distribution, is also known as Cauchy distribution. It also has a relativistic version, the Breit-Wigner distribution f(E) ~ [(E^2-M^2)^2 + M^2\Gamma^2]^-1.
The Fano resonance, characterized by f(E) ~ [(E-M + q*\Gamma)][(E-M)^2 + \Gamma^2]^{-1}, is [E-M+q*\Gamma] times the Lorentzian. Thus the Fano resonance is asymmetric even when \Gamma is E-independent. The parameter q is called the Fano parameter, which represents the ratio between the resonant and direct scattering. In the absence of direct scattering, q-> infinity and the Fano resonance reduces to the Lorentzian.
The Feshbach-Fano partitioning separates the scattering into two parts, the resonance part P, and the direct (background) part Q, with P+Q=1. The operators related to P and Q are defined somewhat arbitrarily, but once they are defined, the total scattering process (defined by a Hamiltonian H), can be unique separated into two parts, PHP, and [QHQ + \Delta(E) - i\Gamma].
