The interaction of an isolated state |Theta> with a known infinite-dimensional quantum system S-infinity(b) containing multiple eigenvalue bands is considered. An example is the interaction of an isolated molecular state with the electromagnetic field. This problem is the main subject of spectroscopy. Another example is the interaction of an isolated molecular state of a molecule situated on the surface of a solid with this solid. This is the subject of the surface state physics. Such problems are usually treated within the formalism of the perturbation expansion. Recently a new mathematical method for the treatment of such problems was suggested. This method produces correct solution of the combined system, however strong the interaction between the state |Theta> and the system S-infinity(b). Key quantities in this method are characteristic functions f(v)(epsilon) and functions omega(v)(epsilon). Functions f(v)(epsilon) are constructed from the quantities describing unperturbed system S-infinity(b) and from the known interaction of this system with the state |Theta>. Functions omega(v)(epsilon) are then expressed in terms of the characteristic functions f(v)(epsilon). An important mathematical problem is the calculation of the functions omega(v)(epsilon), once functions f(v)(epsilon) are known. A general mathematical solution to this problem is obtained for the case when characteristic functions f(v)(epsilon) are polynomials. Since each function which is continuous in a finite interval can be approximated (to any desired degree of accuracy) with some polynomial, this solution presents an important mathematical tool for the practical application of the suggested method. Obtained results are illustrated with few model examples of the interaction of an isolated state with an infinite quantum system.
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
qk(E) for the Fano parameter of the resonance state k is derived that contains the interaction of the state k with neighbored states $l\ensuremath{\ne}k$ 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 $l\ensuremath{\ne}k.$ Under certain conditions, the energy dependent
qk(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
qk(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.