On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators

Abstract

We study spectral and scattering properties of the Laplacian H (σ) = -Δ in L2(ℝ+2) corresponding to the boundary condition ∂u/∂v + σu = 0 for a wide class of periodic functions σ. The Floquet decomposition leads to problems on an unbounded cell which are analyzed in detail. We prove that the wave operators W±(H(σ),H(0)) exist.