Reverse Agmon estimates and nodal intersection bounds in forbidden regions

TL;DR

This paper establishes lower bounds for eigenfunctions in forbidden regions of a Riemannian manifold, complementing classical decay estimates, and applies these results to eigenfunction restriction and nodal intersection bounds.

Contribution

It provides a partial converse to Agmon estimates by deriving exponential lower bounds in forbidden regions under certain mass control conditions.

Findings

Derived exponential lower bounds for eigenfunctions in forbidden regions.

Applied bounds to hypersurface restriction problems.

Obtained nodal intersection estimates in forbidden regions.

Abstract

Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \mathbb{R})$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 \Delta_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in the forbidden region. The point of this note is to prove a partial converse to the Agmon estimates (ie. exponential {\em lower} bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region $\{ V< E \}$ arbitrarily close to the caustic $ \{ V = E \}.$ We then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates.