Logarithmic decays of unstable states

It is known that the survival amplitude of unstable quantum states deviates from exponential relaxations and exhibits decays that depend on the integral and analytic properties of the energy distribution density. In the same scenario, model independent dominant logarithmic decays t−1−α0log t of the survival amplitude are induced over long times by special conditions on the energy distribution density. While the instantaneous decay rate exhibits the dominant long time relaxation 1 /t, the instantaneous energy tends to the minimum value of the energy spectrum with the dominant logarithmic decay 1/(tlog 2t) over long times. Similar logarithmic relaxations have already been found in the dynamics of short range potential systems with even dimensional space or in the Weisskopf-Wigner model of spontaneous emission from a two-level atom. Here, logarithmic decays are obtained as a pure model independent quantum effect in general unstable states