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

TL;DR

This paper develops a dynamical approach to low-frequency scattering in 2D and 3D, deriving explicit formulas for scattering amplitudes and exploring applications in cloaking and solvable models.

Contribution

It extends the transfer matrix and Dyson series framework from 1D to higher dimensions for low-frequency scattering analysis.

Findings

Explicit formulas for low-frequency scattering amplitude

Effective use of the approach in exactly solvable problems

Potential application in low-frequency cloaking

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.