Exponentially small adiabatic invariant for the Schroedinger equation

We study an adiabatic invariant for the time-dependent Schroedinger equation which gives the transition probability across a gap from time t' to time t. When the hamiltonian depends analytically on time, and t'=-∞, t=+∞ we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit. (orig.)