Increase and decrease of tunneling by variation of the barrier height

We show, based on some results from classical analysis, that time-dependent perturbations can either increase or decrease the tunneling probability. The results depend on the form of the function T(√V), where T is the transmission coefficient and V the potential. If T(√V) is convex, the perturbation will always result in a decrease of the tunneling; if T(√V) is concave, both increase as well as decrease occur.