Stable complex-rotation eigenvalues that correspond to no full resonances in scattering: Examples in positron scattering by the helium ion

Hyperspherical close-coupling calculations for S- and P-wave positron scattering by He⁺ ions produce no full resonances in the eigenphase sum δ(E) in the two regions of energy E where stable eigenvalues E_r−iΓ∕2 with large Γ were found previously by the complex-rotation method (CRM); δ(E) increases only by one radian in the lower-E region, and even decreases almost monotonically in the higher-E region, implying time advance, rather than time delay, due to the collision. However, the peaks found in the trace of time-delay matrix, Tr Q(E), are consistent with the CRM eigenvalues. This suggests that these eigenvalues indeed represent S-matrix poles in the complex-E plane, but that their effects on scattering are almost masked by the background δ due to the large Γ. This work uses a general relation Tr Q(E)=2ℏ(dδ∕dE), proved here for any functional form of δ(E), and hence, both on and off resonance. This is a generalization of the well-known single-channel time-delay formula and of the multichannel formula proved previously for the Breit-Wigner resonance with a constant background S matrix.