Rupert L. Frank
On the Laplacian in the halfspace with a periodic boundary condition
(274K, Postscript)
ABSTRACT. We study spectral and scattering properties of the Laplacian $H^{(\sigma)} = -\Delta$ in $L_2(\R^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function $\sigma$. For non-negative $\sigma$ we prove that $H^{(\sigma)}$ 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 $H^{(\sigma)}$ under mild assumptions on $\sigma$.