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.
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Frank, R. On the Laplacian in the halfspace with a periodic boundary condition. Ark Mat 44, 277–298 (2006). https://doi.org/10.1007/s11512-005-0012-3
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DOI: https://doi.org/10.1007/s11512-005-0012-3