Scattering, Homogenization, and Interface Effects for Oscillatory Potentials with Strong Singularities
Vincent Duchêne, Michael I. Weinstein · Multiscale Modeling and Simulation · 2011
We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation [Formula: see text] for [Formula: see text] and [Formula: see text]. Here [Formula: see text] has mean zero and [Formula: see text] as [Formula: see text]. The distorted plane waves of [Formula: see text] are solutions of the form [Formula: see text], [Formula: see text] outgoing as [Formula: see text]. We derive their [Formula: see text] small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, [Formula: see text], follow. Let [Formula: see text] denote the homogenized transmission coefficient associated with the average potential [Formula: see text]. If the potential is smooth, then classical homogenization theory gives asymptotic expansions of, for example, distorted plane waves and transmission and reflection coefficients. Singularities of [Formula: see text] or discontinuities of [Formula: see text] are “interfaces” across which a solution must satisfy interface conditions (continuity or jump conditions). To satisfy these conditions it is necessary to introduce interface correctors, which are highly oscillatory in [Formula: see text]. Our theory admits potentials which have discontinuities in the microstructure, [Formula: see text], as well as strong singularities in the background potential, [Formula: see text]. A consequence of our main results is that [Formula: see text], the error in the homogenized transmission coefficient, is (i) [Formula: see text] if [Formula: see text] is continuous and (ii) [Formula: see text] if [Formula: see text] has discontinuities. Moreover, in the discontinuous case, the correctors are highly oscillatory in [Formula: see text], i.e., [Formula: see text] for [Formula: see text]. Thus a first order corrector is not well defined since [Formula: see text] does not have a limit as [Formula: see text]. This expression may have limits which depend on the particular sequence through which [Formula: see text] tends to zero. The analysis is based on a (preconditioned) Lippman–Schwinger equation, introduced by S.E. Golowich and M.I. Weinstein [Multiscale Model. Simul., 3 (2005), pp. 477–521].