On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering

Rupert L. Frank; Roman G. Shterenberg

doi:10.4171/dm/157

JournalsdmVol. 9pp. 57–77

On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering

Rupert L. Frank
Royal Institute of Technology St. Petersburg State University Department of Mathematics Department of Physics Lindstedtsvägen 25 Ul'yanovskaya 1, Petrodvorets, 10044 Stockholm, Sweden 198504 St. Petersburg, Russia
Roman G. Shterenberg
Royal Institute of Technology St. Petersburg State University Department of Mathematics Department of Physics Lindstedtsvägen 25 Ul'yanovskaya 1, Petrodvorets, 10044 Stockholm, Sweden 198504 St. Petersburg, Russia

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Abstract

We study spectral and scattering properties of the Laplacian $H^{(σ)} = - Δ$ in $L_{2} (R_{+}^{2})$ corresponding to the boundary condition $\frac{\partial u}{\partial ν} + σ u = 0$ for a wide class of periodic functions $σ$ . 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 which are analyzed in detail.

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Rupert L. Frank, Roman G. Shterenberg, On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering. Doc. Math. 9 (2004), pp. 57–77

DOI 10.4171/DM/157