John Toth (McGill University)

Reverse Agmon estimates for Schr\"{o}dinger eigenfunctions

Let $(M,g)$ be a compact, Riemannian manifold and $V in C^{infty}(M; R)$. 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 $h o 0^+.$ 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. Our main result is 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 }.$ I will explain this result in my talk and then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates. This is joint work with Xianchao Wu.