Reverse Agmon 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 Schrödinger operator $P(h) = - h^2 \Delta _g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \rightarrow 0^+.$ The well-known Agmon–Lithner estimates [5] are exponential decay estimates (i.e. 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 reverse Agmon estimate (i.e. exponential lower bound for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowed 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.