Scattering and Localization Properties of Highly Oscillatory Potentials

作者：Duchene V^*; Vukicevic I; Weinstein M I

来源：Communications on Pure and Applied Mathematics, 2014, 67(1): 83-128.

DOI：10.1002/cpa.21459

We investigate scattering, localization, and dispersive time decay properties for the one-dimensional Schrodinger equation with a rapidly oscillating and spatially localized potential q(epsilon) = q(x, x/epsilon), where q(x, y) is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low-energy (k small) behavior of scattering quantities, e. g., the transmission coefficient t(q epsilon)(k) as epsilon tends to zero. We derive an effective potential well sigma(epsilon(x))(eff) = -epsilon(2)Lambda(eff)(x)such that t(q epsilon)(k) - t(sigma effc)(k/t) is small, uniformly for k is an element of R as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if epsilon, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half-plane on the imaginary axis at a distance of order epsilon(2) from 0. It follows that the Schrodinger operator H-q epsilon = -partial derivative(2)(x) + q(epsilon)(x) has an L-2 bound state with negative energy situated a distance O(epsilon(4)) from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: %26lt;br%26gt;vertical bar(1 + vertical bar . vertical bar)(-3)e-itH(qc) P-c psi(0)vertical bar L infinity %26lt;= %26lt;br%26gt;Ct(-1/2)(1 + epsilon(4)(integral(R) Lambda(eff))(2)t)(-1) vertical bar(1 + vertical bar.vertical bar(3))psi(0)vertical bar(L1) %26lt;br%26gt;where P-c denotes the projection onto the continuous spectral part of H-q epsilon.

• 出版日期2014-1