Complex billiards

An analysis is made of the Dirichlet problem for the Helmholtz equation in an infinite region with radiation at infinity. Quantization conditions are formulated which in certain cases yield a quasi-classical asymptotic system of complex eigenvalues representing Green function poles whose imaginary parts are related to radiation losses. For the case of a limited region with an axis of symmetry, these quantum conditions lead to the splitting of the natural frequencies for the interacting symmetric quasi-classical vibrations with nonoverlapping oscillation zones. This construct is based on the complex extension of the phase space of billiards connected with this region and on an analysis of a special kind of Lagrangian submanifolds in this complex.