Transfer matrix in scattering theory: A survey of basic properties and recent developments

TL;DR

This paper provides a comprehensive overview of transfer matrices in one-dimensional scattering theory, including basic properties, recent developments, and applications like unidirectional invisibility and inverse scattering solutions.

Contribution

It introduces a dynamical formulation of scattering theory relating transfer matrices to time-dependent Schrödinger solutions, offering new methods for inverse scattering and device design.

Findings

Derived dynamical equations for reflection and transmission amplitudes

Developed an exact solution for inverse scattering problems

Constructed potentials with prescribed scattering properties

Abstract

We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.