Soluble model of a break-up process

Abstract
The exact solution of a one-dimensional problem, representing the scattering of a particle by another particle which is bound to a fixed centre of force, is given. All the interactions have zero range and are described by boundary conditions. The possible processes are elastic scattering and break-up of the bound system. The elastic scattering and break-up amplitudes are explicitly determined, and the behaviour of the corresponding cross-sections is discussed. At high energies, the incident particle tends to transfer its whole momentum to the bound one, giving rise to a strong peak in the break-up cross-section. The analytic behaviour of the amplitudes is examined. The Riemann surface of the elastic scattering amplitude has three sheets. The break-up threshold gives rise to cubic-root branch points. The remaining singularities are a finite number of poles. The exact amplitudes are compared with those given by the impulse approximation (in first and second order) and by Born’s approximation. It is found that these approximations are reliable only at high incident energies and within the width of the dominant peak of the break-up cross-section.