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Journal of Spectral Theory


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Volume 8, Issue 4, 2018, pp. 1443–1486
DOI: 10.4171/JST/231

Published online: 2018-10-22

Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

Andrea Mantile[1], Andrea Posilicano[2] and Mourad Sini[3]

(1) Université de Reims Champagne-Ardenne, France
(2) Università dell’Insubria, Como, Italy
(3) Austrian Academy of Sciences, Linz, Austria

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 $\delta′$-type, either assigned on $\Gamma$ or on $\Sigma \subset \Gamma$.

Keywords: Limiting absorption principle, scattering matrix, boundary conditions, selfadjoint extensions

Mantile Andrea, Posilicano Andrea, Sini Mourad: Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces. J. Spectr. Theory 8 (2018), 1443-1486. doi: 10.4171/JST/231