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)=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^{iα_n y} 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 mathfrak{z}, the potentials mathfrak{z} sum_n=-infty^inftyδ(x)δ(y-an) and a^-1mathfrak{z}δ(x)\[1+2cos(2πy/a)\] have the same scattering amplitude for a< λleq 2a.