Lower bounds for eigenfunction restrictions in lacunary regions

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 $M\setminus \text{supp}({\pi }_{\ast }\mu )\mathrm{\ne }\mathrm{\varnothing }$ where $\pi :T^{\ast }M\to M$ is the canonical projection. Using Carleman estimates we prove that for any real-smooth closed hypersurface $H\subset (M\setminus \text{supp}({\pi }_{\ast }\mu ))$ sufficiently close to $\text{supp}({\pi }_{\ast }\mu ),$ and for all $\delta >0,$ $${\int }_H\mathrm{\mid }{u}_h{\mathrm{\mid }}^{2}d\sigma \ge C_{\delta }{e}^{-[d(H,\text{supp}({\pi }_{\ast }\mu ))+\delta ]\mathrm{/}h}$$ as $h\to 0^{+}$. We also show that the result holds for eigenfunctions of Schrödinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.