Journal of the American Mathematical Society

Author(s): Nicolas Burq; Maciej Zworski
Journal: J. Amer. Math. Soc. 17 (2004), 443-471.
MSC (2000): Primary 35B37, 35P20, 81Q20
Posted: February 3, 2004
Retrieve article in: PDF DVI PostScript

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.

1.: 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.
2.: C. Bardos, G. Lebeau, and J. Rauch.
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary.
SIAM J. Control Optim. 30:1024-1065, 1992. MR 94b:93067
3.: J.-F. Bony and L. Michel.
Microlocalization of resonant states and estimates of the residue of scattering amplitude,
Comm. Math. Phys., to appear.
4.: N. Burq.
Control for Schrodinger equations on product manifolds.
Unpublished, 1992.
5.: N. Burq.
Contrôle de l'équation des plaques en présence d'obstacles strictement convexes.
Mémoire de la S.M.F., 55, 1993.
Supplément au Bulletin de la Société Mathématique de France. MR 95d:93007
6.: N. Burq.
Semi-classical estimates for the resolvent in non trapping geometries.
Int. Math. Res. Notices 5:221-241, 2002. MR 2002k:81069
7.: N. Burq.
Smoothing effect for Schrödinger boundary value problems.
Preprint, 2002.
8.: N. Burq and P. Gérard,
Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,
Comptes Rendus de L'Académie des Sciences, 749-752, t. 325, Série I, 1996. MR 98j:93052
9.: N. Burq and and G. Lebeau.
Mesures de défaut de compacité, application au système de Lamé,
Ann. Sci. École Norm. Sup. (4), No. 34, 817-870, 2001. MR 2003a:35186
10.: N. Burq and M. Zworski.
Eigenfunctions for partially rectangular billiards.
math.SP/0312098.
11.: P.A. Chinnery and V.F. Humphrey.
Experimental visualization of acoustic resonances within a stadium-shaped cavity.
Physical Review E, 53, 1996, 272-276.
12.: T. Christiansen and M. Zworski.
Resonance wave expansions: two hyperbolic examples. Comm. Math. Phys. 212:323-336, 2000. MR 2001j:58050
13.: Y. Colin de Verdière and B. Parisse.
Équilibre instable en régime semi-classique. I. Concentration microlocale.
Comm. Partial Differential Equations, 9-10, 19, 1535-1563, 1994. MR 96b:58112
14.: N. Dencker, J. Sjöstrand, and M. Zworski.
Pseudospectra of semiclassical (pseudo)differential operators.
Comm. Pure Appl. Math. 57:384-415, 2004.
15.: M. Dimassi and J. Sjöstrand.
Spectral asymptotics in the semiclassical limit.
Cambridge Universtity Press, 1999. MR 2001b:35237
16.: H. Donnelly.
Quantum unique ergodicity.
Proc. Amer. Math. Soc. 131, 2945-2951, 2003.
17.: Ch. Gérard.
Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes.
Mém. Soc. Math. France (N.S.) 31 (1988), 146 pp. MR 91e:35168
18.: Ch. Gérard and J. Sjöstrand.
Resonances en limite semiclassique et exposants de Lyapunov,
Comm. Math. Phys. 116-2, 193-213, 1988. MR 89f:35057
19.: P. Gérard and E. Leichtnam,
Ergodic Properties of Eigenfunctions for the Dirichlet Problem,
Duke Math. Jour. 71, 559-607, 1993. MR 94i:35146
20.: L. Guillopé.
Sur la distribution des longueurs des géodésiques fermées d'une surface compacte à bord totalement géodésique.
Duke Math. J. 53:827-848, 1986. MR 88e:11042
21.: J.P. Françoise and V. Guillemin.
On the period spectrum of a symplectic mapping,
J. Funct. Anal. 100:317-358, 1993. MR 92j:58083
22.: A. Haraux.
Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,
J. Math. Pures Appl. 68-4:457-465, 1989. MR 91e:93039
23.: B. Helffer and J. Sjöstrand.
Resonances en limite semi-classique,
Mémoire de la S.M.F., 114, 1986. MR 88i:81025
24.: L. Hörmander.
The Analysis of Linear Partial Differential Operators. Vol. III, IV.
Springer-Verlag, Berlin, 1985. MR 87d:35002a; MR 87d:35002b
25.: A. Iantchenko, J. Sjöstrand, and M. Zworski.
Birkhoff normal forms in semi-classical inverse problems.
Math. Res. Lett. 9:337-362, 2002. MR 2003f:35284
26.: M. Ikawa.
Decay of solution of the wave equation in the exterior of several convex bodies.
Annales de l'Institut Fourier 38(2):113-146, 1988. MR 90a:35028
27.: S. Jaffard
Contrôle interne exact des vibrations d'une plaque rectangulaire.
Portugal. Math. 47 (1990), no. 4, 423-429. MR 91j:93051
28.: J.P. Kahane.
Pseudo-périodicité et séries de Fourier lacunaires.
Annales Sc. de l'Ecole Normale Supérieure 79, 1962. MR 27:4019
29.: G. Lebeau.
Contrôle de l'équation de Schrödinger.
Journal de Mathématiques Pures et Appl. 71:267-291, 1992. MR 93i:35018
30.: J.L. Lions.
Contrôlabilité exacte, perturbation et stabilisation des systèmes distribués, R.M.A.,
Masson, 1988. MR 90a:49040
31.: E. Lindenstrauss.
Invariant measures and arithmetic quantum unique ergodicity,
preprint, 2003.
32.: R.B. Melrose and J. Sjöstrand.
Singularities of Boundary Value Problems I and II,
Comm. in Pure Appl. Math. 31 and 35, 593-617 and 129-168, 1978 and 1982. MR 58:11859; MR 83h:35120
33.: L. Miller.
How violent are fast controls for Schrödinger equation?
preprint, 2003.
34.: P. Sarnak.
Arithmetic quantum chaos.
In The Schur lectures (1992) (Tel Aviv), volume 8 of Israel Math. Conf. Proc., pages 183-236. Bar-Ilan Univ., Ramat Gan, 1995. MR 96d:11059
35.: J. Sjöstrand.
Geometric bounds on the density of resonances for semiclassical problems,
Duke Math. J. 60:1-57, 1990. MR 91e:35166
36.: J. Sjöstrand.
A trace formula and review of some estimates for resonances.
In Microlocal Analysis and Spectral Theory, volume 490 of NATO ASI series C, pages 377-437. Kluwer, 1997. MR 99e:47064
37.: J. Sjöstrand and M. Zworski.
Complex scaling and the distribution of scattering poles.
Journal of the A.M.S. 4(4):729-769, 1991. MR 92g:35166
38.: J. Sjöstrand and M. Zworski.
Quantum monodromy and semiclassical trace formulæ.
Journal d'Mathématiques Pures et Appl. 81:1-33, 2002.
39.: J.A.K. Suykens and J. Vandewalle (Eds.).
Nonlinear Modeling: advanced black-box techniques,
Kluwer Academic Publishers Boston, June 1998.
40.: S.H. Tang and M. Zworski.
From quasimodes to resonances.
Math. Res. Lett. 5:261-272, 1998. MR 99i:47088
41.: J. Wunsch and M. Zworski.
Distribution of resonances for asymptotically Euclidean manifolds.
J. Diff. Geom. 55:43-82, 2000. MR 2002e:58062
42.: S. Zelditch.
Quantum unique ergodicity.
math-ph/0301035.
43.: S. Zelditch and M. Zworski,
Ergodicity of eigenfunctions for ergodic billiards.
Comm. Math. Phys. 175:673-682, 1996. MR 97a:58193
44.: E. Zuazua.
Contrôlabilité exacte en temps arbitrairement petit de quelques modèles de plaques.
Appendix A.1 to [30]. MR 90a:49040

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35B37, 35P20, 81Q20

Nicolas Burq
Affiliation: Université Paris Sud, Mathématiques, Bât 425, 91405 Orsay Cedex, France
Email: Nicolas.burq@math.u-psud.fr

Maciej Zworski
Affiliation: Mathematics Department, University of California, Evans Hall, Berkeley, California 94720
Email: zworski@math.berkeley.edu

DOI: 10.1090/S0894-0347-04-00452-7
PII: S 0894-0347(04)00452-7
Received by editor(s): May 14, 2003
Posted: February 3, 2004
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS