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

Authors : Ali Mostafazadeh

Pages : 472-527

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Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on thebasic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions ofpotential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jostsolutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We thenoffer a simple derivation of the Lippmann–Schwinger equation and the 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 potentialto the solution of the time-dependent Schrödinger equation for an effective nonunitary two-level quantum system. Weprovide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use itto devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflectionand 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 importantapplications in designing optical devices with a variety of functionalities, amounts to providing an explicit constructionfor a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at anygiven wavenumber.
Keywords : Potential scattering, transfer matrix, complex potential, locally period potential, spectral singularity, tunable unidirectional invisibility, Born approximation, Dyson series, single-mode inverse scattering