Lower bounds for eigenfunction restrictions in lacunary regions

TL;DR

This paper establishes lower bounds for the mass of Laplace eigenfunctions restricted to certain hypersurfaces outside their defect measure support, using Carleman estimates, with applications to Schrödinger eigenfunctions and integrable systems.

Contribution

It provides new exponential lower bounds for eigenfunction restrictions in lacunary regions, extending to Schrödinger operators and integrable quantum systems.

Findings

Lower bounds depend exponentially on the distance from the defect measure support.

Results apply to eigenfunctions of Schrödinger operators.

Applications include eigenfunctions on warped products and QCI systems.

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) \neq \emptyset$ where $\pi:T^*M \to M$ is the canonical projection. Using Carleman estimates we prove that for any real-smooth closed hypersurface $H \subset (M\setminus \text{supp} (\pi_* \mu))$ sufficiently close to $ \text{supp}(\pi_* \mu),$ and for all $\delta >0,$ $$ \int_{H} |u_h|^2 d\sigma \geq C_{\delta}\, e^{- [\, d(H, \text{supp}(\pi_* \mu)) + \,\delta] /h} $$ as $h \to 0^+$. We also show that the result holds for eigenfunctions of Schr\"odinger operators and give applications to eigenfunctions on warped products and joint eigenfunctions of quantum completely integrable (QCI) systems.