Reverse Agmon Estimates in Forbidden Regions

  • John A. TothEmail author
  • Xianchao Wu


Let (Mg) be a compact, Riemannian manifold and \(V \in C^{\infty }(M; {{\mathbb {R}}})\). Given a regular energy level \(E > \min V\), we consider \(L^2\)-normalized eigenfunctions, \(u_h,\) of the Schrödinger operator \(P(h) = - h^2 \Delta _g + V - E(h)\) with \(P(h) u_h = 0\) and \(E(h) = E + o(1)\) as \(h \rightarrow 0^+.\) 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.



We would like to thank Jeff Galkowski and Andreas Knauf for many helpful discussions. We also thank Stéphane Nonnenmacher and the anonymous referees for detailed comments on earlier versions of the paper.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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