Stretched Exponential Decay of a Quasiparticle in a Quantum Dot

The decay of a quasiparticle in an isolated quantum dot is considered. At relatively small time the probability to find the system in the initial state decays exponentially: $P(t)\sim \exp(-\Gamma t)$, in accordance with the golden rule. However, the contributions to $P(t)$ accounting for the discreteness of final three-particle states, five-particle states, etc. decay much slower being $\sim (\Delta_3/\Gamma)^n \exp(-\Gamma t/(2n+1))$ for $2n+1$ final particles. Here $\Delta_3 \ll \Gamma$ is the level spacing for three-particle states available via the direct decay. These corrections are dominant at large enough time and slow down the decay to become $\ln (P)\sim -\sqrt{t}$ asymptotically. $P(t)$ fluctuates strongly in this regime and the analytical formula for the distribution $W(P)$ is found.