One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics
type: Item
instance of: scholarly article
title: One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics
zbMATH Open document ID: 0894.34077
author: R. Nowell
author: Fritz Gesztesy
author: Walter Pötz
published in: Differential and Integral Equations
publication date: 1998-05-17T00:00:00Z
review text: The authors establish a general framework of stationary scattering theory of one-dimensional quantum systems with nontrivial spatial asymptotics. The principal method used consists in combination of the Weyl-Titchmarsh theory of second-order differential operators generated by the differential expression \[ \tau =-{d^2\over dt^2}+ V(t),\quad t\in \mathbb{R}, \] with explicit spectral representation in terms of the generalized eigenfunction expansion for such operators. Using this method, the explicit formulas for the so-called transmission and reflection coefficients \(T^{\pm }(\lambda)\), \(R^{\pm }(\lambda)\) via Weyl \(m\)-functions and normalized real eigenfunctions are obtained as well as the detailed description of the connection between scattering data and spectral representation of the underlying Schrödinger operators. It is also shown that the obtained results have a broad range of application in various (physical) situations, like charge transport in quasi-one-dimensional mesoscopic solid state structure.
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 81U10
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 1153421
zbMATH Keywords: scattering theory
zbMATH Keywords: quantum system
zbMATH Keywords: spectral representation
zbMATH Keywords: Weyl-Titchmarsh theory
zbMATH Keywords: \(m\)-function
reviewed by: Ondřej Došlý
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1153421