# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.333081

Arkiv for Matematik, 44, (2), 277-298, 2004.12
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.
@inproceedings{rupert2004on,
title={On the Laplacian in the halfspace with a periodic boundary condition},
author={Rupert L. Frank},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203617103718890},
booktitle={Arkiv for Matematik},
volume={44},
number={2},
pages={277-298},
year={2004},
}

Rupert L. Frank. On the Laplacian in the halfspace with a periodic boundary condition. 2004. Vol. 44. In Arkiv for Matematik. pp.277-298. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203617103718890.