  In a quantum system with a smoothly and slowly varying Hamiltonian, which
approaches a constant operator at times $t\to \pm \infty$, the transition
probabilities between adiabatic states are exponentially small. They are
characterized by an exponent that depends on a phase integral along a path
around a set of branch points connecting the energy level surfaces in complex
time. Only certain sequences of branch points contribute. We propose that these
sequences are determined by a topological rule involving the Stokes lines
attached to the branch points. Our hypothesis is supported by theoretical
arguments and results of numerical experiments.
