  We study quantum-mechanical tunneling between symmetry-related pairs of
regular phase space regions that are separated by a chaotic layer. We consider
the annular billiard, and use scattering theory to relate the splitting of
quasi-degenerate states quantized on the two regular regions to specific paths
connecting them. The tunneling amplitudes involved are given a semiclassical
interpretation by extending the billiard boundaries to complex space and
generalizing specular reflection to complex rays. We give analytical
expressions for the splittings, and show that the dominant contributions come
from {\em chaos-assisted}\/ paths that tunnel into and out of the chaotic
layer.
