Off-shell T matrices in one, two, and three dimensions

We discuss the calculation of the off-shell T matrix in one, two, and three dimensions using both the Beliaev-Galitskii relation and the inhomogeneous Schroedinger equation. The off-shell T matrix depends, in general, on three quantities: the incoming and outgoing momenta and the energy of the collision, all of which can be chosen independently. We give a simple proof of the fact that for low-momentum scattering the T matrix depends only on the energy of the collision. Using the inhomogeneous Schroedinger equation we derive analytical results for the fully off-shell T matrix in one, two, and three dimensions for hard-sphere central potentials. The usual form of the Beliaev-Galitskii relation is not quite correct for such potentials. We derive the appropriate correction term and show that it corresponds exactly to the contribution to the T matrix from the inhomogeneous term in the inhomogeneous Schroedinger equation