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Reverse Agmon Estimates in Forbidden Regions

  • John A. TothEmail author
  • Xianchao Wu
Article
  • 22 Downloads

Abstract

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.

Notes

Acknowledgements

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.

References

  1. 1.
    Carmona, R., Simon, B.: Pointwise bounds on eigenfunctions and wave packets in \(N\)-body quantum systems. V. Lower bounds and path integrals. Commun. Math. Phys. 80(1), 59–98 (1981)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Canzani, Y., Toth, J.A.: Nodal sets of Schrödinger eigenfunctions in forbidden regions. Ann. Henri Poincaré 17(11), 3063–3087 (2016)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Erdélyi, A.: Asymptotic Expansions, Dover Books in Mathematics. Dover, New York (1955)CrossRefGoogle Scholar
  4. 4.
    Gray, A.: Tubes, Progress in Mathematics, vol. 221. Birkhäuser-Verlag, Basel (2004)Google Scholar
  5. 5.
    Helffer, B.: Semi-classical analysis for the Schrödinger operator and applications. In: Lecture Notes in Mathematics, vol. 1336. Springer, Berlin (1988)CrossRefGoogle Scholar
  6. 6.
    Han, X., Hassell, A., Hezari, H., Zelditch, S.: Completeness of boundary traces of eigenfunctions. Proc. Lond. Math. Soc. 111(3), 749–773 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Helffer, B., Sjöstrand, J.: Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Simon, B.: Semiclassical analysis of low lying eigenvalues. II. Tunneling. Ann. Math. (2) 120(1), 89–118 (1984)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Toth, J.A., Zelditch, S.: Counting nodal lines which touch the boundary of an analytic domain. J. Differ. Geom. 81(3), 649–686 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Toth, J.A., Zelditch, S.: Norms of Modes and Quasimodes Revisited. Contemporary Mathematics, vol. 320. AMS, Providence (2003)zbMATHGoogle Scholar
  11. 11.
    Whitney, H.: Analytic extensions of functions defined in closed sets. Trans. AMS 36(1), 63–89 (1934)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17(4), 661–692 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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