Ray representation of scattering amplitude in the presence of resonances

A new representation of the resonant scattering amplitude over classical paths is found for the problems of scattering at centre-symmetrical potential of a barrier type and in the case of pseudo-crossing of two potential curves. The quasiclassical quantization conditions for the complex poles of the $S$-matrix are derived with the aid of classical paths. It is shown that the most essential contribution of resonances arises when the energy. levels are degenerate. Under certain conditions, the results obtained can be applied for non spherical potentials of a barrier type.