Green functions for generalized point interactions in 1D: A scattering approach

Abstract

Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, requires some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities which may hide important physical aspects of the problem. In this work we present a new method to calculate the exact Green functions for general point interactions in 1D. 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.

Revtex, 9 pages, 3 EPS figures. To be published in PRA

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• The quantum non-linear Schrodinger model with point-like defect
• Green's function approach for quantum graphs: an overview
• Distributional approach to point interactions in one-dimensional quantum mechanics
• Exact Green's function for rectangular potentials and its application to quasi-bound states
• Finite Temperature Quantum Field Theory with Impurities
• Single point potentials with total resonant tunneling
• Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models
• Algebraic approach to multiple defects on the line and application to Casimir force
• Green function approach for scattering quantum walks
• Resonant Transmission in One-Dimensional Quantum Mechanics with Two Independent Point Interactions: Full Parameter Analysis
• Superdiffusivity of quantum walks: A Feynman sum-over-paths description
• The role of scalar current coupling along surfaces
• Duality of bounded and scattering wave systems with local symmetries