Transfer matrix formulation of scattering theory in two and three dimensions

  title={Transfer matrix formulation of scattering theory in two and three dimensions},
  author={Farhang Loran and Ali Mostafazadeh},
  journal={Physical Review A},
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) $. This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than one has been an important open problem for decades. We give a complete solution for this problem and discuss some of its applications. In… 

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
Transfer matrix for long-range potentials
We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials $v(x)$ and derive its basic properties. We outline a dynamical formulation of the
Transfermatrix in scattering theory: a survey of basic properties and recent developments
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
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
Exceptional points and pseudo-Hermiticity in real potential scattering
We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these
Exactness of the Born approximation and broadband unidirectional invisibility in two dimensions
Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on
Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line
We use the transfer matrix formulation of scattering theory in two-dimensions to treat the scattering problem for a potential of the form $v(x,y)=\zeta\,\delta(ax+by)g(bx-ay)$ where $\zeta,a$, and
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
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
Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications
In two dimensions, the standard treatment of the scattering problem for a delta-function potential, v(r) = z δ(r), leads to a logarithmic singularity which is subsequently removed by a


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
Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility
The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the $S$-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that
Perturbative Unidirectional Invisibility
We outline a general perturbative method of evaluating scattering features of finite-range complex potentials and use it to examine complex perturbations of a rectangular barrier potential. In
Quantum scattering in two dimensions
A self‐contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in
Active Invisibility Cloaks in One Dimension
We outline a general method of constructing finite-range cloaking potentials which render a given finite-range real or complex potential $v(x)$ unidirectionally reflectionless or invisible at a
Renormalized contact potential in two dimensions
We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional
Unidirectionally invisible potentials as local building blocks of all scattering potentials
We give a complete solution of the problem of constructing a scattering potential $v(x)$ that possesses scattering properties of one's choice at an arbitrary prescribed wave number. Our solution