Configuration properties of S-matrix and time-delay for potentials decaying like |χ|^−α, α >1

Abstract

The time-delay in a simple quantum mechanical scattering process is a simple mechanical scattering process is defined to be the limit as r→∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x̲) ~ |x̲|^−α (α>1) at infinity, we use a smooth localization function with the tail growing like r ^1−α (δ>0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation.