03-309 Rupert L. Frank

On the spectral analysis and scattering theory of the Laplacian on the halfplane with a periodic perturbation on the boundary (609K, Postscript) Jun 27, 03

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Abstract. We study the spectrum of the Laplacian $H^(\sigma)=-\Delta$ on $L_2(\R^2_+)$ 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 a non-compact cell, which are analyzed in detail. This allows us to prove under the condition $\sigma\geq0$ that $H^(\sigma)$ is unitarily equivalent to the Neumann Laplacian $H^(0)$, the equivalence being provided by the wave operators. In the general case the existence of additional channels of scattering is investigated, which are due to (possibly embedded) eigenvalues of the problems from the Floquet decomposition.

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