The $L^2$ Behavior of Eigenfunctions Near the Glancing Set

TL;DR

This paper investigates how Laplace eigenfunctions behave near the glancing set on a hypersurface, establishing optimal bounds for their restrictions and demonstrating these bounds are sharp, with applications to quantum ergodic restriction theorems.

Contribution

The paper determines the optimal power for which the restriction of eigenfunctions remains bounded near the glancing set and shows these bounds are sharp, advancing understanding of eigenfunction behavior.

Findings

Optimal power s_0 identified for boundedness of eigenfunction restrictions.

Bounds are shown to be sharp using examples on disk and sphere.

Applications to quantum ergodic restriction theorems are provided.

Abstract

Let $M$ be a compact manifold with or without boundary and $H\subset M$ be a smooth, interior hypersurface. We study the restriction of Laplace eigenfunctions solving $(-h^2\Delta_g-1)u=0$ to $H$. In particular, we study the degeneration of $u|_H$ as one microlocally approaches the glancing set by finding the optimal power $s_0$ so that $(1+h^2\Delta_H)_+^{s_0}u|_H$ remains uniformly bounded in $L^2(H)$ as $h\to 0$. Moreover, we show that this bound is saturated at every $h$-dependent scale near glancing using examples on the disk and sphere. We give an application of our estimates to quantum ergodic restriction theorems.