Lower Bounds for Eigenfunction Restrictions in Lacunary Regions

Abstract

Let (M, g) be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $M {\setminus } {{\text {supp}}}\!(\pi _* \mu ) \ne \emptyset$ where $\pi :T^*M \rightarrow M$ is the canonical projection. Using Carleman estimates we prove that for any real smooth closed hypersurface $H \subset (M{\setminus } {{\text {supp}}}\!(\pi _* \mu ))$ sufficienly close to $\text {supp} \, \pi _* \mu$ and for all $\delta >0,$

$$\begin{aligned} \int _{H} |u_h|^2 d\sigma _{_{\!H}} \ge C_{\delta } e^{- [ \, {\varphi (\tau _{_{\!H}})} + \delta \, ] /h}, \end{aligned}$$

as $h \rightarrow 0^+.$ Here, $\varphi (\tau ) = \tau + O(\tau ^2)$ and $\tau _{_{\!H}}:= d(H, {{\text {supp}}}\!(\pi _* \mu ))$. We also show that an analogous result holds for eigenfunctions of Schrödinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.