Transition from discrete to continuous spectrum: separable potential in one dimension

Abstract

An analysis is made of the transition from the discrete to the continuous spectrum for a separable potential in one dimension. The role played by the length of the box and the convergence parameter, ε, in the different limiting operations is discussed. Relations are found between scattering and perturbation theory matrices and wave functions in momentum representation. In particular, the known expression relating the level shift to the phase shift is recovered. The scattering and Brillouin–Wigner perturbation wave functions are in general not simply related by a phase factor.