Exterior mass estimates and $L^2$ restriction bounds for Neumann data along hypersurfaces

TL;DR

This paper establishes new exterior mass estimates and $L^2$ restriction bounds for Laplace eigenfunctions and Neumann data along hypersurfaces on compact Riemannian manifolds, extending to semiclassical Schrödinger eigenfunctions.

Contribution

It introduces novel exterior mass estimates for eigenfunction restrictions and derives uniform $L^2$ bounds for Neumann data along hypersurfaces, applicable to Schrödinger operators.

Findings

Mass estimates for eigenfunction restrictions in exterior regions

O(1) $L^2$ restriction bounds for Neumann data

Applicability to semiclassical Schrödinger eigenfunctions

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.