2018/04/17 by John A. Toth, Toth, John A., Xianchao Wu +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1804.06380
openalex publication_date 2018/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a compact, Riemannian manifold and V ∈ C∞(M; ℝ). Given a regular energy level E > min V, we consider L2-normalized eigenfunctions, uh, of the Schrodinger operator P(h) = - h2 Δg + V - E(h) with P(h) uh = 0 and E(h) = E + o(1) as h → 0+. The well-known Agmon-Lithner estimates \citeHel 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 dE associated with the degenerate Agmon metric (V-E)+ g with support in the forbidden region. The point of this note 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.