On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering
The authors study the Laplacian \[ H^{(\sigma)}u= -\Delta u\quad\text{on }\mathbb{R}^2_+ \] together with a boundary condition of the third type \[ {\partial u\over\partial\nu}+ \sigma u= 0\quad\text{on }\mathbb{R}\times \{0\}, \] where \(\nu\) denotes the exterior unit normal and where the function \(\sigma: \mathbb{R}\to\mathbb{R}\) is assumed to be \(2\pi\)-periodic. The authors prove that for nonnegative \(\sigma\), \(H^{(\sigma)}\) is unitary equivalent to the Neumann-Laplacian \(H^{(0)}\). For part I, cf. ibid. 8, 547--565 (2003; Zbl 1142.35374).
scientific article; zbMATH DE number 2095717