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 $M\setminus \text{supp}({\pi }_{\ast }\mu )\ne \varnothing$ 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 $\delta >0,$ $${\int }_H|u_h{|}^{2}d\sigma \ge C_{\delta }{e}^{-[d(H,\text{supp}({\pi }_{\ast }\mu ))+\delta ]/h}$$ 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.

References in corpus (1)

• Nodal domains of Maass forms I