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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 2024 MCQ for Journal of the American Mathematical Society is 4.83.

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Geometric control in the presence of a black box
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by Nicolas Burq and Maciej Zworski;
J. Amer. Math. Soc. 17 (2004), 443-471
DOI: https://doi.org/10.1090/S0894-0347-04-00452-7
Published electronically: February 3, 2004
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Abstract:

We apply the “black box scattering” point of view to problems in control theory for the Schrödinger equation and in high energy eigenvalue scarring. We show how resolvent bounds with origins in scattering theory, combined with semi-classical propagation, give quantitative control estimates. We also show how they imply control for time dependent problems.
References
    BS A. Bäcker, R. Schubert, and P. Stifter. On the number of bouncing ball modes in billiards. J. Phys. A: Math. Gen. 30:6783-6795, 1997.
  • Claude Bardos, Gilles Lebeau, and Jeffrey Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065. MR 1178650, DOI 10.1137/0330055
    • BM J.-F. Bony and L. Michel. Microlocalization of resonant states and estimates of the residue of scattering amplitude, Comm. Math. Phys., to appear. Bu92 N. Burq. Control for Schrodinger equations on product manifolds. Unpublished, 1992.
  • Nicolas Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 55 (1993), 126 (French, with English and French summaries). MR 1254820
  • Nicolas Burq, Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not. 5 (2002), 221–241. MR 1876933, DOI 10.1155/S1073792802103059
    • Bu02 N. Burq. Smoothing effect for Schrödinger boundary value problems. Preprint, 2002.
  • Nicolas Burq and Patrick Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 7, 749–752 (French, with English and French summaries). MR 1483711, DOI 10.1016/S0764-4442(97)80053-5
  • Nicolas Burq and Gilles Lebeau, Mesures de défaut de compacité, application au système de Lamé, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 6, 817–870 (French, with English and French summaries). MR 1872422, DOI 10.1016/S0012-9593(01)01078-3
    • BZ3 N. Burq and M. Zworski. Eigenfunctions for partially rectangular billiards. arXiv:math.SP/0312098. CH96 P.A. Chinnery and V.F. Humphrey. Experimental visualization of acoustic resonances within a stadium-shaped cavity. Physical Review E, 53, 1996, 272-276.
  • T. Christiansen and M. Zworski, Resonance wave expansions: two hyperbolic examples, Comm. Math. Phys. 212 (2000), no. 2, 323–336. MR 1772249, DOI 10.1007/s002200000211
  • Yves Colin de Verdière and Bernard Parisse, Équilibre instable en régime semi-classique. I. Concentration microlocale, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1535–1563 (French, with French summary). MR 1294470, DOI 10.1080/03605309408821063
    • DSZ N. Dencker, J. Sjöstrand, and M. Zworski. Pseudospectra of semiclassical (pseudo)differential operators. Comm. Pure Appl. Math. 57:384–415, 2004.
  • Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654, DOI 10.1017/CBO9780511662195
    • Do H. Donnelly. Quantum unique ergodicity. Proc. Amer. Math. Soc. 131, 2945–2951, 2003.
  • C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 31 (1988), 146 (French, with English summary). MR 998698
  • C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), no. 2, 193–213 (French, with English summary). MR 939046
  • Patrick Gérard and Éric Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), no. 2, 559–607. MR 1233448, DOI 10.1215/S0012-7094-93-07122-0
  • Laurent Guillopé, Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique, Duke Math. J. 53 (1986), no. 3, 827–848 (French). MR 860674, DOI 10.1215/S0012-7094-86-05345-7
  • J.-P. Françoise and V. Guillemin, On the period spectrum of a symplectic mapping, J. Funct. Anal. 100 (1991), no. 2, 317–358. MR 1125229, DOI 10.1016/0022-1236(91)90114-K
  • A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl. (9) 68 (1989), no. 4, 457–465 (1990) (French, with English summary). MR 1046761
  • B. Helffer and J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) 24-25 (1986), iv+228 (French, with English summary). MR 871788
  • Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 87
  • A. Iantchenko, J. Sjöstrand, and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett. 9 (2002), no. 2-3, 337–362. MR 1909649, DOI 10.4310/MRL.2002.v9.n3.a9
  • Mitsuru Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 113–146 (English, with French summary). MR 949013, DOI 10.5802/aif.1137
  • S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire, Portugal. Math. 47 (1990), no. 4, 423–429 (French, with English summary). MR 1090480
  • Jean-Pierre Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. (3) 79 (1962), 93–150 (French). MR 154060
  • G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9) 71 (1992), no. 3, 267–291 (French, with English summary). MR 1172452
  • J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
    • Li03 E. Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity, preprint, 2003.
  • Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
    • Mi03 L. Miller. How violent are fast controls for Schrödinger equation? preprint, 2003.
  • Peter Sarnak, Arithmetic quantum chaos, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 183–236. MR 1321639
  • Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. MR 1047116, DOI 10.1215/S0012-7094-90-06001-6
  • J. Sjöstrand, A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 377–437. MR 1451399
  • Johannes Sjöstrand and Maciej Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), no. 4, 729–769. MR 1115789, DOI 10.1090/S0894-0347-1991-1115789-9
    • SjZw02 J. Sjöstrand and M. Zworski. Quantum monodromy and semiclassical trace formulæ. Journal d’Mathématiques Pures et Appl. 81:1-33, 2002. BB J.A.K. Suykens and J. Vandewalle (Eds.). Nonlinear Modeling: advanced black-box techniques, Kluwer Academic Publishers Boston, June 1998.
  • Siu-Hung Tang and Maciej Zworski, From quasimodes to resonances, Math. Res. Lett. 5 (1998), no. 3, 261–272. MR 1637824, DOI 10.4310/MRL.1998.v5.n3.a1
  • Jared Wunsch and Maciej Zworski, Distribution of resonances for asymptotically Euclidean manifolds, J. Differential Geom. 55 (2000), no. 1, 43–82. MR 1849026
    • Ze S. Zelditch. Quantum unique ergodicity. arXiv:math-ph/0301035.
  • Steven Zelditch and Maciej Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), no. 3, 673–682. MR 1372814
  • J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
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Bibliographic Information
  • Nicolas Burq
  • Affiliation: Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex, France
  • MR Author ID: 315457
  • Email: Nicolas.burq@math.u-psud.fr
  • Maciej Zworski
  • Affiliation: Mathematics Department, University of California, Evans Hall, Berkeley, California 94720
  • MR Author ID: 227055
  • Email: zworski@math.berkeley.edu
  • Received by editor(s): May 14, 2003
  • Published electronically: February 3, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 443-471
  • MSC (2000): Primary 35B37, 35P20, 81Q20
  • DOI: https://doi.org/10.1090/S0894-0347-04-00452-7
  • MathSciNet review: 2051618