Mathematical model of resonances and tunneling in system with bound states

The asymptotics of deduction in the pole of the scattering matrix analytical prolongation by the pole imaginary part inclination to zero is studied for the case, when the space phase of the quantum-mechanical system constitutes the direct sum of two spaces; the operator of the unperturbed evolution leads each of these spaces and in one of them it has the discrete spectrum and in the other - the continuous one. The perturbation operator mixes the subspaces and generates the resonance. It is proved that in such a system by accomplishing certain symmetry conditions the scattering amplitude will sharply change in the vicinity of the scattering matrix pole actual part and the tunneling or the scattering matrix amplitude resonance will be observed in the system