Decay rates of resonance states at high level density

The time-dependent Schr\"odinger equation of an open quantum-mechanical system is solved by using the stationary biorthogonal eigenfunctions of the non-Hermitian time-independent Hamilton operator. We calculate the decay rates at low and high level densities in two different formalism. The rates are generally time dependent and oscillate around an average value due to the nonorthogonality of the wave functions. The decay law is studied disregarding the oscillations. In the one-channel case, it is proportional to t^-b with b3/2 in all cases considered, including the critical region of overlapping where the nonorthogonality of the wave functions is large. Starting from the shell model, we get b2 for two and four open decay channels and all coupling strengths to the continuum. When the closed system is described by a random matrix, b1+K/2 forK= 2 and 4 channels. This law holds in a limited time interval. The distribution of the widths is different in the two models when more than one channel is open. This leads to the different exponents b in the power law. Our calculations are performed with 190 and 130 states, respectively, most of them in the critical region. The theoretical results should be proven experimentally by measuring the time behavior of deexcitation of a realistic quantum system. 1996 The American Physical Society.