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 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-i omega t psi (r), with psi(r) satisfying the Helmholtz equation, [del 2 + k2 epsilon(r; k)]psi(r) = 0; omega and k being, respectively, the angular frequency and wave number of the incident wave; and epsilon(r; k) denoting the relative permittivity of the carrier medium, which, in general, takes complex values. We obtain explicit formulas for lowfrequency 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.