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

The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator of an effective nonunitary 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 in which 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 ${e}^{\ensuremath{-}i\ensuremath{\omega}t}\ensuremath{\psi}(\mathbf{r})$, with $\ensuremath{\psi}(\mathbf{r})$ satisfying the Helmholtz equation, $[{\ensuremath{\nabla}}^{2}+{k}^{2}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varepsilon}}(\mathbf{r};k)]\ensuremath{\psi}(\mathbf{r})=0$; $\ensuremath{\omega}$ and $k$ being, respectively, the angular frequency and wave number of the incident wave; and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varepsilon}}(\mathbf{r};k)$ 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.