The generalized point interaction in one dimension

Sergio A Albeverio, Friedrich Gesztesy, Raphael J. Høegh-Krohn +1 more

Taksu Cheon, Tamás Fülöp, Izumi Tsutsui

Alexandre G. M. Schmidt, Bin Kang Cheng, M. G. E. da Luz

Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.

Pavel Exner, Harald Grosse

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.

Sergio A Albeverio, Friedrich Gesztesy, Raphael J. Høegh-Krohn +1 more

W.T.M. Caspers