Reverse Agmon Estimates in Forbidden Regions

Abstract

Let (M, g) be a compact, Riemannian manifold and V ∈ C ∞ (M ; R). Given a regular energy level E > min V , we consider L 2 -normalized eigenfunctions, u h , of the Schrödinger operator P (h) = −h 2 ∆ g + V − E(h) with P (h)u h = 0 and E(h) = E + o(1) as h → 0 + . The well-known Agmon-Lithner estimates [Hel88] 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.The point of this note is to prove a reverse Agmon estimate (ie. 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.

Improved Lower Bound for Analytic Schrödinger Eigenfunctions in Forbidden Regions

“…In [TW20] ((M, g, V ) is only required to be smooth), under the control and monotonicity assumptions (see [TW20, Definitions 1 and 2]), by applying Carleman estimate to pass across the caustic hypersurface [TW20, Theorem 1] authors prove that for any ε > 0 and h ∈ (0, h 0 (ε)],…”

“…where β(ε) = O(ε) as ε → 0 + , A(δ 1 , δ 2 ) ⊂ Ω E is an annular domain near the boundary (precise definitions refer to [TW20]) and constant τ 0 ≥ 1. As authors pointed out in [TW20], (1.5) is a partial reverse Agmon estimate, our objective in this paper is to get an improved result in the case where (M, g, V ) is analytic. This is precisely the point of inequality (1.7).…”