Scattering, homogenization, and interface effects for oscillatory potentials with strong singularities (Q2905613)

The paper deals with the scattering problem for the one-dimensional Schrödinger equation with a decaying potential which is the sum of a slowly varying part and a rapidly oscillating part, \(q_\epsilon=q(x,x/\epsilon)\), \(\epsilon \ll 1\). The goal is to derive an expansion (in small \(\epsilon\)) of the distorted plane wave and of the transmission coefficient, and to give error bounds. The emphasis is made on the effects of singularities of the potential: the slowly varying part is assumed to have smooth and singular components, while the rapidly varying part may have discontinuities. The singularities require interface correctors to an expansion derived by the classical method of multiple scales. These correctors are related to the asymptotics of boundary layers arising in works on homogenization of divergence form operators on bounded domains. It is shown, in particular, that in the case of discontinuous \(q_\epsilon\), the difference between the transmission coefficient and the transmission coefficient associated with the averaged potential is of order \(\epsilon\) but involve highly oscillating terms having, in general, no limit as \(\epsilon\to 0\).