Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces. (English) Zbl 1402.35194
Summary: We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts \(\Sigma\) of) compact hypersurfaces \(\Gamma= \partial \Omega\), \(\Omega \subset \mathbb R^n\). For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space \(\mathbb R^n\). Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, \(\delta\) and \(\delta^\prime\)-type, either assigned on \(\Gamma\) or on \(\Sigma \subset \Gamma\).
MSC:
| 35P25 | Scattering theory for PDEs |
| 35J15 | Second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
limiting absorption principle; scattering matrix; boundary conditions; selfadjoint extensionsReferences:
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