# Reverse Agmon Estimates and some applications

Seminar in Microlocal Analysis and Mathematical PhysicsTitle：Reverse Agmon Estimates and some applications  Speaker：Dr Xianchao WU, McGill Univ, CanadaTime：2018.September 24-28 at 4pmLocation: Center for Mathematical Sciences, Room 813Abstract: 2018.September 24 4pmLet $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \R)$. Given a regular energy level $E > \min V$, we consi...

Seminar in Microlocal Analysis and Mathematical Physics

Title：Reverse Agmon Estimates and some applications

Speaker：Dr Xianchao WU, McGill Univ, Canada

Time：2018.September 24-28 at 4pm

Location: Center for Mathematical Sciences, Room 813

Abstract:

2018.September 24 4pm

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 $h \to 0^+.$ The well-known Agmon-Lithner estimates 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.

2018.September 25 4pm

The point of this talk 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 causti$be a compact, Riemannian manifold and$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 \to 0^+.$The well-known Agmon-Lithner estimates are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region$\{ 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$with support in the forbidden region. 2018.September 26 4pm The point of this talk is to prove a partial converse to the Agmon estimates (ie. exponential {\em lower} bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region$\{ 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.

2018.September 27 4pm

We consider L^2-normalized eigenfunctions of the semiclassical Schrodinger operator on a compact manifold. The well-known Agmon-Lithner estimates are exponential decay estimates (ie. upper bounds)  for  eigenfunctions in the forbidden region. The decay rate is given in terms of the Agmon distance function which is associated with the degenerate Agmon metric with support in the forbidden region.

2018.September 28 4pm

The point of this talk is to prove a partial converse to the Agmon estimates (ie. exponential  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 arbitrarily close to its boundary.

We then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates.