Theory of Resonant States: An Exact Analytical Approach for Open Quantum Systems

Time-domain resonances and the ultimate fate of a decaying quantum state

“…The above considerations have led to a formulation of the time evolution of decay in terms of a purely discrete expansion involving the residues at the poles of the outgoing Green's function to the problem. These residues are proportional to the resonant states of the system that satisfy the Schrödinger equation with outgoing boundary conditions [16,17,29].…”

“…For propagation along the external region of the potential it is convenient to write the outgoing Green's function as [16,17] G + (r,r ; k) = G + (a,r ; k)e ik(r−a) , r < a, r a.…”

“…Substitution of (6) into (5) and the resulting expression into (4) allows us to express the time-dependent decaying solution as the resonant expansion [16,17] (…”

“…This implies that the so-called resonant states cannot be normalized in the usual sense and also that the usual rules concerning orthogonality and completeness do not apply. The above old developments, however, have evolved over the years into a consistent theoretical framework which is particularly useful for the description of time-dependent problems [16,17]. It may be proved that resonant states may be obtained also from the residues at the complex poles of the outgoing Green's function to the problem which also provide the normalization rule for these states.…”

“…It is worth emphasizing that in this approach one only needs to know the resonant states along the internal region and at the boundary value of the interaction potential. The time dependence of the decaying solution is given as a pole expansion [16,17] in which each term is given by the product of an expansion coefficient, a resonant state, and a Moshinsky function [18]. Each expansion coefficient represents the overlap between the initial state and the corresponding resonant state.…”