  We study viscoelastic subdiffusion in bistable and periodic potentials within
the Generalized Langevin Equation approach. Our results justify the (ultra)slow
fluctuating rate view of the corresponding bistable non-Markovian dynamics
which displays bursting and anti-correlation of the residence times in two
potential wells. The transition kinetics is asymptotically
stretched-exponential when the potential barrier $V_0$ several times exceeds
thermal energy $k_BT$ ($V_0\sim 2 - 10 k_BT$) and it cannot be described by the
non-Markovian rate theory (NMRT). The well-known NMRT result approximates,
however, ever better with the increasing barrier height, the most probable
logarithm of the residence times. Moreover, the rate description is gradually
restored when the barrier height exceeds a fuzzy borderline which depends on
the power law exponent of free subdiffusion $\alpha$. Such a potential-free
subdiffusion is ergodic. Surprisingly, in periodic potentials it is not
sensitive to the barrier height in the long time asymptotic limit. However, the
transient to this asymptotic regime is extremally slow and it does profoundly
depend on the barrier height. The time-scale of such subdiffusion can exceed
the mean residence time in a potential well, or in a finite spatial domain by
many orders of magnitude. All these features are in sharp contrast with an
alternative subdiffusion mechanism involving jumps among traps with the
divergent mean residence time in these traps.
