Decay rates of resonance states at high level density

Abstract
The time dependent Schrödinger equation of an open quantum mechanical system is solved by using the stationary bi-orthogonal eigenfunctions of the non-Hermitean time independent Hamilton operator. We calculate the decay rates at low and high level density in two different formalism. The rates are, generally, time dependent and oscillate around an average value due to the non-orthogonality of the wavefunctions. The decay law is studied disregarding the oscillations. In the one-channel case, it is proportional to t^-b with b ≈ 3/2 in all cases considered, including the critical region of overlapping where the non-orthogonality of the wavefunctions is large. Starting from the shell model, we get b ≈ 2 for 2 and 4 Open decay channels and all coupling strengths to the continuum. When the closed system is described by a random matrix, b ≈ 1 + K/2 for K = 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 are 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 behaviour of de-excitation of a realistic quantum system.