Reverse Agmon Estimates in Forbidden Regions

• Published in: Annales Henri Poincaré 2020, Vol.21 (1), p.303-325
• Main Authors: Toth, John A., Wu, Xianchao
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Decay rate; Dynamical Systems and Ergodic Theory; Eigenvectors; Elementary Particles; Energy levels; Estimates; Hyperspaces; Lower bounds; Mathematical and Computational Physics; Mathematical Methods in Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Riemann manifold; Theoretical; Upper bounds
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-019-00867-3

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 Δ 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 [ 5 ] are exponential decay estimates (i.e. 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 (i.e. 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.