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: $P\left(t\right)\mathrm{}\sim \exp \left(-\Gamma t\right),$ in accordance with the golden rule. However, the contributions to $P\left(t\right)\mathrm{}$ accounting for the discreteness of final three-particle states, five-particle states, etc. decay much slower being $\sim ({\Delta }_3/\Gamma {)}^n\exp [-\Gamma t/(2n+1)\mathrm{}]$ 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 $\ln \mathrm{}\left(P\right)\mathrm{}\sim -\sqrt{t}$ asymptotically. $P\left(t\right)\mathrm{}$ fluctuates strongly in this regime and the analytical formula for the distribution $W\left(P\right)\mathrm{}$ is found.