Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

Abstract

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) = ζ δ(ax + by)g(bx − ay) where ζ, a, and b are constants, δ(x) is the Dirac δ function, and g is a real-or complex-valued function. We map this problem to that of v(x, y) = ζ δ(x)g(y) and give its exact and analytic solution for the following choices of g(y): i) A linear combination of δ-functions, in which case v(x, y) is a finite linear array of two-dimensional δ-functions; ii) A linear combination of e iαny with α n real; iii) A general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of two-dimensional δ-functions. We also prove a general theorem that gives a sufficient condition for different choices of g(y) to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and z, the potentials z ∞ n=−∞ δ(x)δ(y − an) and a −1 zδ(x)[1 + 2 cos(2πy/a)] have the same scattering amplitude for a < λ ≤ 2a.

Exactness of the Born approximation and broadband unidirectional invisibility in two dimensions

“…v l (x, y) = e iαy γe −i2αx w − (x, y) + e iαx w + (x, y) , (35) v r (x, y) = e iαy e −iαx w − (x, y) + γe 2iαx w + (x, y) , (36) where γ = 0, 1. Then, according to Theorem 2, for k ∈ (α/ √ 2, α], v l is unidirectionally left-invisible, if…”

“…Now, consider the w − (x, y) of (39) with g 1 (x) and g 2 (y) given by n 1 = n 2 = 1, and let a := L 1 and b := L 2 . Substituting this choice for w − (x, y) in (36) and setting γ = 0 yield the potential (20) with z := −k 2 z − .…”

Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects

“…The application of the transfer-matrix method for the double-delta-function potential [25], which does not involve divergent terms, yields (38) with H…”

“…Refs. [24,25] outline an alternative solution of the scattering problem for (5) that avoids the divergences of the standard approaches. Following the strategy pursued in Ref.…”

Potentials with identical scattering properties below a critical energy

“…Details of the application of this approach for solving specific scattering problems are given in Refs. [13,15,18].…”