Theory of decaying states based on a method of coupled equations: Kapur-Peierls and Siegert resonant states and the 'extended' Hellmann-Feynman theorem

A theoretical treatment of long-living decaying states in resonance phenomena is given from the viewpoint of decaying bound states. The effective Hamiltonian for the Kapur-Peierls resonant state is constructed in this theory by distinct potentials or operators. This theory is based on a method of coupled equations similar to the Faddeev equation, and hence no projection operators are needed to construct the effective Hamiltonian. The 'extended' Hellmann-Feynman theorem applied to the resonant states is developed. Particularly, the energy gradient of the Kapur-Peierls eigenvalue is found in a compact form in terms of the original bound state which decays and the reaction operator. The difference between the Siegert resonant-state energy and the particular solution of the Kapur-Peierls eigenvalue problem at the resonant-state energy is estimated. A simple example of tunnelling problems is given.