Functional Analysis and Scattering Theory

We formulate the nonrelativistic scattering problem as an integral equation with a kernel which is completely continuous for all energies. We then are able to give a rigorous justification for the Fredholm method, quasiparticle method, and, for weak enough interactions, the Born expansion. We also give an explicit lower bound for the radius of convergence of the Born series and of the Born series modified by the introduction of quasiparticles. We furthermore show that all these expansions coverage uniformly in the physical region of energy and momentum transfer.