Time delay in quantum scattering

As is well known, the knowledge of the scattering cross section and its angular dependence, as a function of energy, is insufficient to determine the phase shifts uniquely. This led Eisenbud and Wigner to propose the measurement of the scattering lifetime or time delay as an additional independent datum. A rigorous time-dependent study of time delay within the framework of Hilbert space formalism is presented. Specifically, Martin's theory of time delay and the validity of the Eisenbud-Wigner time delay formula are extended to spherically symmetric potentials satisfying the asymptotic fall-off rate V(r) → O(r/sup -2-epsilon/). This extension is obtained by use of a maximal estimate of the rate of convergence of the asymptotic condition and the elimination of Martin's requirement that the scattering operator S be three times differentiable with respect to the free-particle Hamiltonian H0. Also presented are related results on the total time a quantum particle spends inside some bounded regions in position space. It is then proved that any two free particles having identical distributions of energy and angular momentum take exactly identical expectation values for the transit time across an arbitrary spherical region centered at the origin in position space. Ways to extend this result to nonfree Hamiltonians are indicated. Finally, the relationship between the position operator and the Eisenbud-Wigner time delay operator is examined. It is shown that the usual method of calculating time delay based on the classical analysis of the position operator is not exact