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 $H^{(\sigma )}=-\Delta$ in $L_2({\mathbb{R}}_{+}^2)$ corresponding to the boundary condition $\frac{\partial u}{\partial \nu }+\sigma u=0$ for a wide class of periodic functions $\sigma$. The Floquet decomposition leads to problems on an unbounded cell which are analyzed in detail. We prove that the wave operators $W_{\pm }(H^{(\sigma )},H^{(0)})$ exist.