On the Laplacian in the halfspace with a periodic boundary condition

Abstract

We study spectral and scattering properties of the Laplacian H^(σ)=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that H^(σ) is unitarily equivalent to the Neumann Laplacian H⁽⁰⁾. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of H^(σ) under mild assumptions on σ.