Exponential decay and geometric aspect of transition probabilities in the adiabatic limit

We consider a quantum mechanical system whose hamiltonian is a time-dependent analytic 11 x n matrix. For n = 2 we establish a generalization of Dykhne formula which gives the transition probability from one energy level to the other in the adiabatic limit. We discuss in particular the geometric nature of this formula. In the general case. n > 2, we prove an upper bound for the probability of such transitions which shows that they are exponentially small. ( 1991 Academic Press. Inc