Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves

TL;DR

This paper improves $L^2$ restriction bounds for Neumann data of Laplace eigenfunctions along closed curves by analyzing their concentration and microlocalization properties.

Contribution

It provides new bounds on Neumann data restriction norms and detailed microlocal analysis techniques for eigenfunction concentration.

Findings

Neumann data $L^2$ norms tend to zero for tangential concentration.

Detailed microlocal analysis of Neumann data away from cotangential directions.

Enhanced understanding of eigenfunction boundary behavior along curves.

Abstract

We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $\gamma$ be a closed smooth curve with unit exterior normal $\nu$. We can show that $\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $\gamma$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $\gamma$ of the Neumann data $h\partial_\nu u_h$ when mircolocalized away the cotangential direction.