PRECISE EXPONENTIAL ESTIMATES IN ADIABATIC THEORY

General adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip. and whose spectrum satisfies a gap condition are studied. An explicit rate of exponential decay is given, which is related to simple geometric quantities associated to the spectrum of the Hamiltonian. for the transition probability between the two parts of the spectrum when the evolution is taken from -infinity to +infinity.