Time delay and the density of states

For the incoming wave, this stationary phase condition is d(—kr — Et/h)/dk — 0, from which we know the motion of the wave packet 

In the absence of any interaction, the outgoing wave packet comes out to the asymptotic region without delay. When scattering occurs, the wave packet spends some time in the potential region before coming out from there if At is positive. If a resonance occurs, the time delay is calculable from Eqs. (15) and (19) as 

The resonance phase shift Sr produces much time delay of Lorentzian shape if T is small. The peak value of the time delay at E = E, is 4h/ r = 4r if the background term is neglected. The Lorentzian time delay averaged over the probability density p(E) is hf p2(E)dE = 2r, of which r is spent for the formation of the QBS and the other r for its decay. 

In a resonance phenomenon in two-body collisions A+B, time delay occurs in the relative A — B motion. The particles A and B stay near each other for much longer time than in the absence of any interactions between them, or form a temporary complex or a QBS with a lifetime r — h/T. 

Sometimes in the literature, a resonance refers to the passage of the phase shift across 7t/2, but that does not necessarily mean steep slope of 5(E), a large time delay, or formation of a QBS. On the other hand, a large time delay is not necessarily associated with the formation of a QBS. For example, Wigner s threshold law for a short-range potential states that the s-wave phase shift 50(E) is proportional to E1/2 at low energies E [33]. This leads to the time delay At a E 1/2, which becomes extremely large at very low E (and is positive if the interaction is attractive), in spite of its irrelevance to any QBS [34], Note 

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