Some properties of the one-dimensional generalized point interactions (a torso)

TL;DR

This paper explores the spectral, scattering, and topological properties of one-dimensional generalized point interactions, introducing two parametrizations and analyzing their implications for models like the Kronig-Penney, including Berry phase effects.

Contribution

It introduces two natural parametrizations for GPI and analyzes their spectral, scattering, and topological properties, including Berry phase and high-energy band behavior.

Findings

GPI exhibits a non-trivial Berry phase.

Three types of high-energy band behavior identified.

Resolved spectral and scattering characteristics of GPI.

Abstract

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.