Eigenvalues of the time-delay matrix in overlapping resonances

We discuss the properties of the lifetime or the time-delay matrix Q(E) for multichannel scattering, which is related to the scattering matrix S(E) by Q = iS(dS†/dE). For two overlapping resonances occurring at energies Eν with widths Γν(ν = 1, 2), with an energy-independent background, only two eigenvalues of Q(E) are proved to be different from zero and to show typical avoided-crossing behaviour. These eigenvalues are expressible in terms of the four resonance parameters (Eν, Γν) and a parameter representing the strength of the interaction of the resonances. An example of the strong and weak interaction in an overlapping double resonance is presented for the positronium negative ion. When more than two resonances overlap (ν = 1, ..., N), no simple representation of each eigenvalue has been found. However, the formula for the trace of the Q-matrix leads to the expression δ(E) = −∑arctan[(Γν/2)/(E − Eν)] + δb(E) for the eigenphase sum δ(E) and the background eigenphase sum δb(E), in agreement with the known form of the state density. The formulae presented in this paper are useful in a parameter fitting of overlapping resonances.