Reverse Agmon Estimates in Forbidden Regions

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 \(h \rightarrow 0^+.\) 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.