Improved Lower Bound for Analytic Schr\"odinger Eigenfunctions in Forbidden Regions

The point of this paper is to improve the reverse Agmon estimate discussed in \cite{TW} with assuming that the Schrodinger operator $P(h) = - h^2 \Delta_g + V - E(h)$, $E(h)\to E$ as $h\to 0^+$, is analytic on a compact, real-analytic Riemannian manifold $(M,g)$. In this paper, by considering a Neumann problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we can prove an improved reverse Agmon estimate on a hypersurface.