Exact Solution of the Two-Dimensional Scattering Problem for a Class of $\delta$-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 $v(x,y)=\zeta \delta (ax+by)g(bx-ay)$ where $\zeta ,a$, and $b$ are constants, $\delta (x)$ is the Dirac $\delta$ function, and $g$ is a real- or complex-valued function. We map this problem to that of $v(x,y)=\zeta \delta (x)g(y)$ and give its exact and analytic solution for the following choices of $g(y)$: i) A linear combination of $\delta$-functions, in which case $v(x,y)$ is a finite linear array of two-dimensional $\delta$-functions; ii) A linear combination of $e^{i{\alpha }_{n}y}$ with ${\alpha }_n$ 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 $\delta$-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 $\lambda$. For example, we show that for arbitrary real and complex parameters, $a$ and $\mathfrak{z}$, the potentials $\mathfrak{z}{\sum }_{n=-\infty }^{\infty }\delta (x)\delta (y-an)$ and $a^{-1}\mathfrak{z}\delta (x)[1+2\cos (2\pi y/a)]$ have the same scattering amplitude for $a<\lambda \le 2a$.

15 pages

References in corpus (4)

• The transfer matrix: a geometrical perspective
• Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions
• Perfect Broadband Invisibility in Isotropic Media with Gain and Loss
• Unidirectional Invisibility and Nonreciprocal Transmission in Two and Three Dimensions

Cited by in corpus (9)

• Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
• Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension
• Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions
• Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions
• Transmission of low-energy scalar waves through a traversable wormhole
• Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects
• Potentials with Identical Scattering Properties Below a Critical Energy
• Scattering by a collection of $\delta$-function point and parallel line defects in two dimensions
• Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution