Exact Solution of the Two-Dimensional Scattering Problem for a Class of $\delta$-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 $b$ 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 $a< \lambda\leq 2a$.