Possibility for time-scale-invariant relaxation in tunneling

We analyze relaxation of a disturbance of the wave function induced by a slowly varying perturbation localized inside a potential barrier. The perturbation is assumed to decay in time exponentially. We demonstrate that for a ''flat'' barrier, whose curvature at the highest point is zero (e.g., a box-shaped barrier), the relaxation is time scale invariant, being proportional to a certain power of time. If the curvature does not vanish but is small enough, the temporal scaling will be observed as an intermediate asymptotic regime, while at very large times the relaxation becomes exponential with a time scale determined solely by the small curvature. Dissipation slows down the relaxation, leaving it time scale invariant.