Additivity of traversal time in multichannel systems

Prove that, for a general multichannel quantum scattering system with short-range interactions, the total two-component traversal time τ_E (defined via the transmission matrix t(E) and reflection amplitudes r_{ii}(E) as τ_E = d/dE ln det[t(E)] + Σ_i [2 i μ_i L/k_i + μ_i r_{ii}(E) e^{2 i k_i L}/k_i^2]) is additive across channels, i.e., τ_E = Σ_i τ_E^{(ii)}, where each channel-resolved traversal time is given by τ_E^{(ii)} = (μ_i/k_i) ∂/∂k_i ln[t_{ii}(E) e^{2 i k_i L}] + μ_i r_{ii}(E) e^{2 i k_i L}/k_i^2, with k_i the channel momentum, μ_i the channel reduced mass, and L half the length of the scattering region.

Background

The paper generalizes the concept of tunneling (traversal) time to coupled-channel systems and defines a total two-component traversal time τE in terms of the determinant of the multichannel transmission matrix t(E) and boundary corrections involving reflection amplitudes. For each channel i, a channel-resolved traversal time τ_E^{(ii)} is derived from the diagonal elements of the Green’s function and expressed using t{ii}(E), r_{ii}(E), and the channel kinematics.

Based on these definitions, the authors assert that the total traversal time should be an additive quantity obtained by summing over all input modes (channels), i.e., τ_E = Σ_i τ_E^{(ii)}. However, they explicitly note that an analogous theorem establishing this additivity has not been proved for multichannel systems, thereby identifying a concrete theoretical gap.