On the quantisation of homoclinic motion

The density of states of a classically chaotic system can be represented as a sum over its periodic orbits. An unstable periodic orbit may be the limit of homoclinic orbits, that in turn accumulate infinite families of satellite periodic orbits with arbitrarily long periods. The Birkhoff-Moser theorem, guaranteeing the convergence of normal forms in a neighbourhood of the homoclinic orbits, is the basis of explicit formulae for the actions of the satellite orbits with large periods. The conclusion of a study of the combined contribution of the satellite orbits to the periodic orbit sum is that homoclinic motion can support a quantum state if the central periodic orbit is quantised by Bohr-Sommerfeld rules and its Lyapunov exponent alpha is sufficiently small. The periodic orbits included in the sum range over the entire homoclinic region, but the wave intensity for the state that they support displays a heavy scar mainly along the central periodic orbit.