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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 $h \to0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decayestimates (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 theforbidden 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 Agmondistance in the forbidden region under a control assumption on eigenfunctionmass in the allowable region $\{ V< E \}$ arbitrarily close to the caustic $ \{V = E \}.$ We then give some applications to hypersurface restriction boundsfor eigenfunctions in the forbidden region along with corresponding nodalintersection estimates.

Reverse Agmon estimates and nodal intersection bounds in forbidden regions