Restriction Bounds for the Free Resolvent and Resonances in Lossy Scattering

Abstract

We establish high energy $L^2$ estimates for the restriction of the free Green's function to hypersurfaces in ${\mathbb{R}}^d$. As an application, we estimate the size of a logarithmic resonance free region for scattering by potentials of the form $V\otimes {\delta }_{\Gamma }$, where $\Gamma \subset {\mathbb{R}}^d$ is a finite union of compact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for solutions to the wave equation with such a potential.

24 pages

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