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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Control of eigenfunctions on surfaces of variable curvature
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by Semyon Dyatlov, Long Jin and Stéphane Nonnenmacher
J. Amer. Math. Soc. 35 (2022), 361-465
Published electronically: August 16, 2021


We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature.

The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the $C^{1+}$ regularity of the stable/unstable foliations.

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Bibliographic Information
  • Semyon Dyatlov
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720; and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 830509
  • ORCID: 0000-0002-6594-7604
  • Email:
  • Long Jin
  • Affiliation: Yau Mathematical Sciences Center, Tsinghua University, People’s Republic of China
  • MR Author ID: 1076590
  • Email:
  • Stéphane Nonnenmacher
  • Affiliation: Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay cedex, France
  • ORCID: 0000-0001-5989-9362
  • Email:
  • Received by editor(s): August 22, 2019
  • Received by editor(s) in revised form: January 27, 2021
  • Published electronically: August 16, 2021
  • Additional Notes: The first author was supported by Clay Research Fellowship, Sloan Fellowship, and NSF CAREER Grant DMS-1749858. The second author was supported by Recruitment Program of Young Overseas Talent Plan. The third author was partially supported by the Agence National de la Recherche, through the grants GERASIC-ANR-13-BS01-0007-02 and ISDEEC-ANR-16-CE40-0013. This project was initiated during the IAS Emerging Topics Workshop on Quantum Chaos and Fractal Uncertainty Principle in October 2017
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 35 (2022), 361-465
  • MSC (2020): Primary 58J51
  • DOI:
  • MathSciNet review: 4374954