On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators

We study spectral and scattering properties of the Laplacian $H^{(\sigma)} = -\Delta$ in $L_2(\R^2_+)$ 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 an unbounded cell which are analyzed in detail. We prove that the wave operators $W_\pm(H^{(\sigma)},H^{(0)})$ exist.

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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