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

TL;DR

This paper improves the reverse Agmon estimate for Schrödinger eigenfunctions in forbidden regions on analytic manifolds by using Neumann problems, Poisson representation, and exterior mass estimates.

Contribution

It introduces an enhanced lower bound for eigenfunctions in forbidden regions by leveraging analyticity and advanced PDE techniques, extending previous results.

Findings

Improved lower bounds for eigenfunctions in forbidden regions.

Application of Poisson representation and exterior mass estimates.

Enhanced reverse Agmon estimates on hypersurfaces.

Abstract

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.