Dynamical formulation of low-frequency scattering in two and three dimensions

Abstract

The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator for an effective non-unitary quantum system. In particular, it admits a Dyson series expansion which turns out to facilitate the construction of the low-frequency series expansion of the scattering data. In two and three dimensions, there is a similar formulation of stationary scattering where the scattering properties of the scatterer are extracted from the evolution operator for a corresponding effective quantum system. We explore the utility of this approach to scattering theory in the study of the scattering of low-frequency time-harmonic scalar waves, $e^{-i\omega t}\psi(\mathbf{r})$, with $\psi(\mathbf{r})$ satisfying the Helmholtz equation, $[\nabla^2+k^2\hat\varepsilon(\mathbf{r};k)]\psi(\mathbf{r})=0$, $\omega$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat\varepsilon(\mathbf{r};k)$ denoting the relative permittivity of the carrier medium which in general takes complex values. We obtain explicit formulas for low-frequency scattering amplitude, examine their effectiveness in the study of a class of exactly solvable scattering problems, and outline their application in devising a low-frequency cloaking scheme.