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

  title={Transfermatrix in scattering theory: a survey of basic properties and recent developments},
  author={Ali Mostafazadeh},
  • A. Mostafazadeh
  • Published 22 September 2020
  • Physics, Mathematics
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… 

Figures from this paper

Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
We offer a consistent dynamical formulation of stationary scattering in two and three dimensions that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear
Dynamical formulation of low-energy scattering in one dimension
The transfer matrix M of a short-range potential may be expressed in terms of the timeevolution operator for an effective two-level quantum system with a time-dependent nonHermitian Hamiltonian. This
Compatibility of transport effects in non-Hermitian nonreciprocal systems
Based on a general transport theory for non-reciprocal non-Hermitian systems and a topological model that encompasses a wide range of previously studied examples, we (i) provide conditions for
Low-frequency scattering defined by the Helmholtz equation in one dimension
The Helmholtz equation in one dimension, which describes the propagation of electromagnetic waves in effectively one-dimensional systems, is equivalent to the time-independent Schrödinger equation.


Scattering Theory and PT-Symmetry
We outline a global approach to scattering theory in one dimension that allows for the description of a large class of scattering systems and their P-, T -, and PT -symmetries. In particular, we
Transfer matrices as nonunitary S matrices, multimode unidirectional invisibility, and perturbative inverse scattering
We show that in one dimension the transfer matrix M of any scattering potential v coincides with the S-matrix of an associated time-dependent non-Hermitian 2 x 2 matrix Hamiltonian H(\tau). If v is
A Dynamical Formulation of One-Dimensional Scattering Theory and Its Applications in Optics
We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are
Scattering theory of waves and particles
Much progress has been made in scattering theory since the publication of the first edition of this book fifteen years ago, and it is time to update it. Needless to say, it was impossible to
Mathematical scattering theory
The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator. There are two different trends in scattering theory for
Nonlinear scattering and its transfer matrix formulation in one dimension
Abstract.We present a systematic formulation of the scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition
Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions
We develop a transfer-matrix formulation of the scattering of electromagnetic waves by a general isotropic medium which makes use of a notion of electromagnetic transfer matrix $\mathbf{M}$ that does
The transfer matrix: A geometrical perspective
Abstract We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as
Transfer matrix formulation of scattering theory in two and three dimensions
In one dimension one can dissect a scattering potential $ v(x) $ into pieces $ v_i(x) $ and use the notion of the transfer matrix to determine the scattering content of $ v(x) $ from that of $ v_i(x)
Variational Principles for Scattering Processes. I
A systematic treatment is presented of the application of variational principles to the quantum theory of scattering. Starting from the time-dependent theory, a pair of variational principles is