#### DOCUMENTA MATHEMATICA, Vol. 9 (2004), 57-77

Rupert L. Frank and Roman G. Shterenberg

On the Scattering Theory of the Laplacian with a Periodic Boundary Condition. II. Additional Channels of Scattering

We study spectral and scattering properties of the Laplacian $H^{(\sigma)} = -\Delta$ in $L_2(\R^2_+)$ corresponding to the boundary condition $\frac{ tial u}{ tial\nu} + \sigma u = 0$ for a wide class of periodic functions $\sigma$. 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 which are analyzed in detail.

2000 Mathematics Subject Classification: Primary 35J10; Secondary 35J25, 35P05, 35P25.

Keywords and Phrases: Scattering theory, periodic operator, Schrödinger operator, singular potential.

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