Anomalous Decay of an Unstable State Coupled with Singular Continuous States: Weak-Coupling Limit

The decay of an unstable quantum state interacting with an environment with a purely singular continuous energy spectrum is studied. Within the framework of the weak coupling Friedrichs-Fano (Newns-Anderson) model, a scaling approach is developed by assuming the local Holder continuity of the environmental integrated density of states (IDOS). It reduces to the conventional van Hove scaling when IDOS is differentiable at the resonant energy. When the Holder index at the resonant energy is less than unity, the survival amplitude is shown to be aM ittag-Lef fl er function of the scaled time, which describes monotonic decay, decay with one local maximum, oscillatory convergence towards a finite value. Then, these predictions are compared with the numerical solution of the integral equation for the survival amplitude and are shown to well reproduce the numerical observations provided that the differential/Holder coefficients are estimated in an appropriate coarse-grained way.