Exterior Mass Estimates and L2-Restriction Bounds for Neumann Data Along Hypersurfaces

Abstract

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $\phi_h$, $(h^2 \Delta - 1)\phi_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^{\delta}$-scales ($0\leq \delta < 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schr\odinger operators.

Locations

• arXiv (Cornell University) - View - PDF
• Carolina Digital Repository (University of North Carolina at Chapel Hill) - View - PDF