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

Abstract: We seek to improve the restriction bounds of Neumann data of semiclassical Schr\"{o}dinger eigenfunctions $u_h$ considered by Christianson-Hassell-Toth \cite{CHT} and Tacy \cite{Tacy2} by studying the $L^2$ restriction bounds of eigenfunctions and their $L^2$ concentration as measured by defect measures. Let $\Gamma$ be a smooth hypersurface with unit exterior normal $\nu$. Our main result says that $| h \partial_\nu u_{h} |_{L^2(\Gamma)}=o(1)$ when ${u_h}$ is admissible.