Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

TL;DR

This paper establishes a limiting absorption principle and constructs generalized eigenfunctions and scattering matrices for Laplace operators with various boundary conditions on hypersurfaces, using operator-valued Weyl functions and Krein-type formulas.

Contribution

It provides a unified framework for analyzing Laplace operators with boundary conditions on hypersurfaces, including explicit formulas for eigenfunctions and scattering matrices.

Findings

Valid for Dirichlet, Neumann, Robin, δ, and δ' boundary conditions

Expresses scattering objects in terms of Weyl functions

Applies Krein-type formulas to relate resolvents

Abstract

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 $\Sigma\subset\Gamma$.