Complete separation of resonance and nonresonance channel spaces

The concept of 'resonance channel space' is proposed. Resonance states can decay into any open channels of common symmetry. They can also decay into any of the eigenchannels defined by diagonalizing the S matrix. However, an isolated resonance is known to decay only into a particular one of the eigenchannels (referred to as Q-eigenchannels) defined by diagonalizing the time-delay matrx Q=iℎSd†/dE, where the dagger indicates the Hermitian conjugate. This particular Q-eigenchannel corresponds to the eigenvalue of Lorentzian shape as a function of the energy E. A theorem generalizing this for N overlapping resonances is proved; only N eigenvalues of the Q matrix are nonzero and exhibit N Lorentzian profiles, avoiding each other around their crossing points, provided that the background S matrix is independent of E. Any of the overlapping resonances can decay only into the N Q-eigenchannels corresponding to these eigenvalues, and not to any other Q-eigenchannels. Thus, the set of Q-eigenchannels can be divided into 'resonance channel space', i.e. the subset of N Q-eigenchannels associated with the resonances and its complement completely irrelevant to the resonances asymptotically. (fast track communication)