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.