A~scattering problem in laminar media

A. L. Piatnitski

Sbornik. Mathematics, Tome 43 (1982) no. 3, pp. 427-441

Résumé

The scattering problem in a laminar medium $$ \Delta u(x)+k^2q(x_1,\dots,x_n,x_1/\varepsilon)u(x)=0 $$ with a radiation condition at infinity is considered. The potential $q(x,y)$ is periodic in the variable $y$. Here $k$ is a large parameter, and $\varepsilon$ is a small parameter with $k\sim\varepsilon^{-\alpha}$, $\alpha>1$. In this paper a formal asymptotic expansion of the solution of this problem is found. To construct it an operator analogous to the canonical Maslov operator is used which acts on a certain Lagrangian manifold not depending on $\varepsilon$. An analogous problem for the Schrödinger equation in a laminar medium is solved. Bibliography: 10 titles.