4. Direct Scattering II

In this chapter the scattering process is described physically and mathematically, and the definition of the scattering operator is provided in terms of the wave operators introduced by Møller. The role of the limiting absorption principle is indicated, and it is shown how the Hamiltonian and its resolvent are related to the potential and the two boundary matrices describing the general self-adjoint boundary condition. The generalized Fourier maps associated with the absolutely continuous spectrum are introduced and their basic properties are outlined. It is shown how the wave operators are related to the generalized Fourier maps and their adjoints and hence how the scattering operator is related to the generalized Fourier maps. It is shown that the scattering matrix defined in terms of the Jost matrix coincides with the scattering matrix derived from the scattering operator. Various other topics are considered such as the properties of the spectral shift function, trace formulas of Buslaev–Faddeev type, and a Bargmann–Birman–Schwinger bound on the number of bound states.