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.