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

TL;DR

The paper addresses improving $L^2$ restriction bounds for Neumann data along a closed curve by linking these bounds to defect measures of Laplace eigenfunctions. It develops a microlocal framework, including a projector $B_{\varepsilon,\delta}$, to separate cotangential and non-cotangential contributions, and proves a first main bound on the microlocalized Neumann data. Under the assumption that the defect measure is tangentially concentrated, it then uses a Rellich identity and defect-measure analysis to obtain an $o(1)$ bound for $\| h \partial_\nu u_h \|_{L^2(\gamma)}$, strengthening prior results. The paper also provides concrete examples on the torus and with Gaussian beams to illustrate sharpness and limitations, confirming the role of diffuseness in enabling improved restriction bounds. Overall, it clarifies when Neumann flux along a curve must vanish in the high-frequency limit and ties semiclassical concentration phenomena to boundary restriction behavior.

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 $γ$ be a closed smooth curve with unit exterior normal $ν$. We can show that $\| h \partial_νu_{h} \|_{L^2(Γ)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $γ$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $γ$ of the Neumann data $h\partial_νu_h$ when mircolocalized away the cotangential direction.

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Definition 2
• Theorem 3
• Remark 2
• Remark 3
• Remark 4
• Lemma 3.1: CGT Lemma 6
• Remark 5
• Lemma 3.2: CGT Lemma 7