Fano resonances in the overlapping regime

The line shape of resonances in the overlapping regime is studied by using the eigenvalues and eigenfunctions of the effective Hamiltonian of an open quantum system. A generalized expression q₊ (E) for the Fano parameter of the resonance state k is derived that contains the interaction of the state k with neighbored states l via the continuum. It is energy dependent since the coupling coefficients between the state k and the continuum show a resonancelike behavior at the energies of the neighbored states l. Under certain conditions, the energy dependent q₊ (E) are equivalent to the generalized complex energy independent Fano parameters that are introduced by Kobayashi et al. in analyzing experimental data. Long-lived states appear mostly isolated from one another in the cross section, also when they are overlapped by short-lived resonance states. The q₊ (E) of narrow resonances allow therefore to study the complicated interplay between different time scales in the regime of overlapping resonance states by controlling them as a function of an external parameter.