Coupling of open and closed channels in Feshbach resonance model

Question

Feshbach resonance is described with coupling of 2 systems differing in the form of potentials :- one is said to produce a bound state (in 'closed' channel) and other is to produce scattering states (in 'open' channel). The two channels are said to be coupled by another potential term. The resonance occurs where a bound state energy from the closed channel matches a scattering state energy in the open channel.

I understand that if we have a 2*2 potential matrix, we may have non-diagonal terms which can be said to bring the coupling between two systems. The calculation is done by taking the schrodinger equation for the whole system and projecting onto the two separate systems using projection operators onto the subspaces of states of the subsystems. Is there a way to describe the resonance (bound state energy = scattering state energy) without dividing the original system into two coupled systems of closed and open channels i.e. taking the schrodinger equation for the whole system with the 2*2 potential containing non-diagonal coupling terms ?

Refer section 2.3 here :- http://www.physics.ox.ac.uk/Users/godun/FilingCabinet/FeshbachTheory.pdf

Answer 1

Sometimes it is possible to treat the problem without a priori splitting it into a coupled set. But in the end something similar does come up. Here I am referring to the complex dilatation method and its spin-offs (www.chem.ucla.edu/~akg/annurev.pc.33.100182.001255.pdf ). Basically it consists of analytically continuing a specific unitary transformation with the effect that the continuous spectrum of the Hamiltonian is moved away from the real axis thus exposing the resonances as isolated poles of its resolvent. The method works beautifully for Coulomb systems such as the Helium atom. But in general it may be hard to find a suitable analytic continuation.