O ct 1 99 9 Some properties of the one - dimensional generalized point interactions ( a torso )

This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.