Scattering resonances of microstructures and homogenization theory

Scattering resonances are the eigenvalues and corresponding eigenmodes which solve the Schrodinger equation H psi = E psi for a Hamiltonian, H, subject to the condition of outgoing radiation at infinity. We consider the scattering resonance problem for potentials which are rapidly varying in space and are not necessarily small in a pointwise sense. Such potentials are of interest in many applications in quantum, electromagnetic, and acoustic scattering, where the environment consists of microstructure, e. g., rapidly varying material properties. Of particular interest in applications are high contrast microstructures, e. g., large pointwise variations of material properties. We develop a perturbation theory for the scattering resonances and eigenvalues of such high contrast and oscillatory potentials. The expansion is proved to be convergent in a norm which encodes the degree of oscillation in the microstructure. Next, we consider the concrete example of two-dimensional microstructure potentials. These correspond, for example, to a class of photonic waveguides with transverse microstructures. The leading order behavior is given by the scattering resonances of a suitable averaged potential, as predicted by classical homogenization theory. We show that the next term in the expansion agrees with that given by a higher order homogenization multiple-scale expansion, with an error term determined by the regularity of the potential. The higher order corrections, which take into account the detailed microstructure, have been shown by the authors to be important for efficient and accurate numerical computation of radiation rates.