Feshbach–Fano partitioning
In
In general, the partitioning formalism is based on the definition of two complementary
- P + Q = 1.
The subspaces onto which P and Q project are sets of states obeying the continuum and the bound state
The projectors P and Q are not defined within the Feshbach–Fano method. This is its major power as well as its major weakness. On the one hand, this makes the method very general and, on the other hand, it introduces some arbitrariness which is difficult to control. Some authors define first the P space as an approximation to the background scattering but most authors define first the Q space as an approximation to the resonance. This step relies always on some physical intuition which is not easy to quantify. In practice P or Q should be chosen such that the resulting background scattering phase or cross-section is slowly depending on the scattering energy in the neighbourhood of the resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical form, it is possible to generate a set of equations defining P and Q on a less arbitrary basis.
The aim of the Feshbach–Fano method is to solve the
whose dimension is equal to the number of interacting resonances and depends parametrically on the scattering energy E. The resonance parameters and are obtained by solving the so-called implicit equation
for z in the lower complex plane. The solution
is the resonance pole. If is close to the real axis it gives rise to a Breit–Wigner or a Fano profile in the corresponding cross section. Both resulting T matrices have to be added in order to obtain the T matrix corresponding to the full scattering problem :