THE LEVEL SPLITTING DISTRIBUTION IN CHAOS-ASSISTED TUNNELING

A compound tunnelling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of p hase space was recently introduced to explain peculiar features of tun nelling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunnelling. We study its consequences for the distribu tion of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunnelling is the only operative mechanism. We have checked that the analytical f orm we obtain agrees with splitting distributions calculated numerical ly for a model system in which chaos-assisted tunnelling is known to b e the dominant mechanism. The distribution depends on two parameters. The first gives the scale of the splittings and is related to the magn itude of the classically forbidden processes, the second gives a measu re of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter param eter is irrelevant; otherwise it sets an energy scale at which the spl itting distribution crosses over from one type of behaviour to another . The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunnelling.