Time dependence, complex scaling, and the calculation of resonances in many‐electron systems

The theory of this paper deals with certain aspects of the formal properties of atomic and molecular highly excited nonstationary states and the problem of calculating their wave functions, energies, and widths. The conceptual framework is a decay theory based on the consistent definition and calculation of the t = 0 localized state, |Ψ. Given this framework, the following topics are treated: (a) The variational calculation of Ψ 0 and E 0 using a previously published theory that generalized the projection operator approach to many‐electron systems. (b) The exact definition of the resonance energy. (c) The possibility of bound states in the continuum. (d) The relation of Ψ 0 to the resonance (Gamow) function Ψ and of the Hamiltonian to the rotated Hamiltonian H (θ) based on the notion of perturbation of boundary conditions in the asymptotic region. (e) The variational calculation of real and complex energies employing matrix elements of H and H 2 with square‐integrable and resonance functions. (f) The mathematical structure of the time evolution of |Ψ 0 〉 and the possibility of observing nonexponential decays in certain autoionizing states that are very close to the ionization threshold. (g) A many‐body theory of atomic and molecular resonances that employs the coordinate rotation method.