SYSTEMATICS OF THE AVERAGE RESONANCE WIDTHS IN OVERLAPPING RESONANCE-SCATTERING AND ITS CONNECTION WITH THE RRKM THEORY

Decay processes of densely distributed quasi-bound states are studied numerically by randomly generating the Hamiltonian matrices. The average decay rate obtained from the Feshbach theory of resonance scattering exhibits systematic behavior against the average density of states (rho), the number of continua (K) and the average coupling strength to the continua (upsilon). The distribution of the decay rates bifurcates into long-lived and short-lived branches when rho is larger than a certain critical value rho(c), which is found to be roughly equal to the inverse of the 0-th order partial width [gamma(part.)]. Thus one can clearly distinguish the isolated resonance regime in the region rho < rho(c) and overlapping resonance regime in the region rho > rho(c). The states belonging to the short-lived branch exhibit a very broad energy spectrum and are recognized as background continua. They are not quasi-bound states in practice. The decay rates of the long-lived branch, on the other hand, systematically decrease with rho at rho much-greater-than rho(c). The average of these decay rates is proportional to [gamma(part.)]-1K(rho)-2. When the short-lived branch is excluded, the average decay rate, [GAMMA/HBAR], roughly agrees with that of the RRKM rate in the region rho almost-equal-to rho(c), where the spectral profile becomes most diffuse. Outside the region of rho almost-equal-to rho(c), [GAMMA/HBAR] is always smaller than the RRKM rate. The above observational is confirmed also by a square-well potential model and ascertains the conventional belief that the RRKM theory holds only when resonances overlap and that it gives the upper bound. It is noteworthy that this RRKM regime corresponds to the critical overlap, rho[gamma(part.)] almost-equal-to 1.