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Geometric Control of Eigenfunctions of Schrödinger Operators

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Research in PDEs and Related Fields

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Abstract

We review the role of the Geometric Control Condition in establishing the observability property from an open set for solutions to the wave, Schrödinger, and eigenfunction equations. We show how to construct surfaces of revolution for which the observability property holds under strictly weaker conditions on the observation set than their counterparts for the wave and Schrödinger equations. We also introduce a class of Schrödinger operators on the two-dimensional sphere for which observability for eigenfunctions holds provided the observation region intersects only three fixed geodesics on the sphere, which only depend on the potential.

The author acknowledges the support of Ministerio de Ciencia, Innovación y Universidades of the Spanish government through grant MTM2017-85934-C3-3-P

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References

  1. N. Anantharaman, F. Macià, Semiclassical measures for the Schrödinger equation on the torus. J. Eur. Math. Soc. (JEMS) 16(6), 1253–1288 (2014)

    Google Scholar 

  2. N. Anantharaman, C. Fermanian-Kammerer, F. Macià, Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures. Am. J. Math. 137(3), 577–638 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. N. Anantharaman, M. Léautaud, F. Macià, Winger measures and observability for the Schrödinger equation on the disk. Invent. Math. 206(2), 485–599 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Azagra, F. Macià, Concentration of symmetric eigenfunctions. Nonlinear Anal. 73(3), 683–688 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Bardos, G. Lebeau, J. Rauch, Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Rend. Sem. Mat. Univ. Politec. Torino 46, 11–31 (1989). 1988. Nonlinear hyperbolic equations in applied sciences

    Google Scholar 

  6. A.L. Besse, Manifolds all of whose geodesics are closed, in Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93 (Springer, Berlin, 1978). With appendices by D.B.A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger, J.L. Kazdan

    Google Scholar 

  7. N. Burq, P. Gérard, Stabilization of wave equations on the torus with rough dampings. Pure Appl. Anal. 2(3), 627–658 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques. Comment. Math. Helv. 54(3), 508–522 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, in London Mathematical Society Lecture Note Series, vol. 268 (Cambridge University Press, Cambridge, 1999)

    MATH  Google Scholar 

  10. J.J. Duistermaat, V.W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Dyatlov, M. Zworski, Mathematical theory of scattering resonances, in Graduate Studies in Mathematics, vol. 200 (American Mathematical Society, Providence, RI, 2019)

    MATH  Google Scholar 

  12. C. Fermanian-Kammerer, Opérateurs pseudo-différentiels semi-classiques, in Chaos en mécanique quantique (Éc. Polytech., Palaiseau, 2014), pp. 53–100

    Google Scholar 

  13. P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire Équations aux dérivées partielles (Polytechnique), pp. 1–19, (1990)–(1991)

    Google Scholar 

  14. P. Gérard, É. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Guillemin, The Radon transform on Zoll surfaces. Adv. Math. 22(1), 85–119 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Humbert, Y. Privat, E. Trélat, Quantum Limits on product manifolds. arXiv:2202.04379 (2022)

    Google Scholar 

  17. J.-i. Itoh, R. Sinclair, Thaw: a tool for approximating cut loci on a triangulation of a surface. Exp. Math. 13(3), 309–325 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire. Portugal. Math. 47(4), 423–429 (1990)

    MathSciNet  MATH  Google Scholar 

  19. V. Komornik, On the exact internal controllability of a Petrowsky system. Journal de Mathématiques Pures et Appliquées. Neuvième Série 71(4), 331–342 (1992)

    MathSciNet  MATH  Google Scholar 

  20. G. Lebeau, Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. (9) 71(3), 267–291 (1992)

    Google Scholar 

  21. G. Lebeau, Équation des ondes amorties, in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Mathematical Physics Studies, vol. 19 (Kluwer Academic Publishers, Dordrecht, 1996), pp. 73–109

    Google Scholar 

  22. P.-L. Lions, T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9(3), 553–618 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Macià, Some remarks on quantum limits on Zoll manifolds. Commun. Partial Differential Equations 33(4–6), 1137–1146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Macià, High-frequency propagation for the Schrödinger equation on the torus. J. Funct. Anal. 258(3), 933–955 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Macià, The Schrödinger flow on a compact manifold: High-frequency dynamics and dispersion, in Modern Aspects of the Theory of Partial Differential Equations. Operator Theory: Advances and Applications, vol. 216 (Springer, Basel, 2011), pp. 275–289

    Google Scholar 

  26. F. Macià, High-frequency dynamics for the Schrödinger equation, with applications to dispersion and observability, in Nonlinear optical and atomic systems. Lecture Notes in Mathematics, vol. 2146 (Springer, Cham, 2015), pp. 275–335

    Google Scholar 

  27. F. Macià, G. Rivière, Concentration and non-concentration for the Schrödinger evolution on Zoll manifolds. Commun. Math. Phys. 345(3), 1019–1054 (2016)

    Article  MATH  Google Scholar 

  28. F. Macià, G. Rivière, Observability and quantum limits for the Schrödinger equation on \(\mathbb {B}{S}^d\), in Probabilistic methods in geometry, topology and spectral theory. Contemporary Mathematics, vol. 739 (American Mathematical Society, Providence, RI, 2019), pp. 139–153

    Google Scholar 

  29. F. Macià, Observability results related to fractional Schrödinger operators. Vietnam J. Math. 49(3), 919–936 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Martinez, An introduction to semiclassical and microlocal analysis (Springer, New York, 2002)

    Book  MATH  Google Scholar 

  31. J. Rauch, M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds. Commun. Pure Appl. Math. 28(4), 501–523 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Zelditch, Maximally degenerate Laplacians. Ann. Inst. Fourier (Grenoble) 46(2), 547–587 (1996)

    Google Scholar 

  34. S. Zelditch, Fine structure of Zoll spectra. J. Funct. Anal. 143(2), 415–460 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. H. Zhu, Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM Control Optim. Calc. Var. 24(4), 1759–1788 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Zworski, Semiclassical analysis, in Graduate Studies in Mathematics, vol. 138 (American Mathematical Society, Providence, RI, 2012)

    Book  Google Scholar 

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Authors and Affiliations

  1. M2ASAI, Universidad Politécnica de Madrid, ETSI Navales, Madrid, Spain

    Fabricio Macià

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  1. Fabricio Macià
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Correspondence to Fabricio Macià .

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  1. Department of Mathematics, University of Monastir, Monastir, Tunisia

    Kaïs Ammari

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Macià, F. (2022). Geometric Control of Eigenfunctions of Schrödinger Operators. In: Ammari, K. (eds) Research in PDEs and Related Fields. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-14268-0_5

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