Scattering by periodic surfaces

Abstract

If Ω⊆ Rν (ν?2) is an exterior domain containing a half-space and contained in a half space, we prove that the wave operators W± = s-limt→±∞ exp(itH) Jexp(−itH0) are partial isometries and that the invariance principle W±(φ) = W± holds for suitable real functions φ on R (’’admissible’’ functions). Here H0 is the negative (distributional) Laplacian in L2(Rν) (ν?2); H is HD or HN , the negative Dirichlet or Neumann Laplacians in ℋ = L2(Ω), respectively; J is an appropriate identification operator; and W± (φ) are defined as were W± , but with H0 and H replaced by φ(H0) and φ(H), respectively. Suppose, in addition, that Ω has a suitable periodicity property and that when H = HN it has a certain mild local compactness property. Then we prove: (1) that W± are asymptotically complete, in the sense that Ran W± = ℋscatt(H) = ℋ○ℋsurf(H) ; (2) that ℋscatt(φ(H)) = ℋscatt(H) and ℋsurf (φ(H)) = ℋsurf(H) for each ’’admissible’’ function φ, and hence that W± (φ) are asymptotically complete in a similar sense for each such φ. Here, for any self-adjoint operator A in ℋ, ℋscatt(A), and ℋsurf(A) are naturally defined, mutually orthogonal subspaces of scattering and surface states of ℋ, respectively. No smoothness assumptions on ∂Ω are made in this paper. Its results entail the asymptotic completeness, in a physically very satisfactory sense, of wave operators describing acoustic and certain types of electromagnetic scattering by a very wide class of periodic surfaces.