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Séminaire Laurent Schwartz — EDP et applications

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San Vũ Ngọc
Spectral invariants for coupled spin-oscillators
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exp. No. 7, 18 p., doi: 10.5802/slsedp.5
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Résumé - Abstract

This text deals with inverse spectral theory in a semiclassical setting. Given a quantum system, the haunting question is “What interesting quantities can be discovered on the spectrum that can help to characterize the system ?” The general framework will be semiclassical analysis, and the issue is to recover the classical dynamics from the quantum spectrum. The coupling of a spin and an oscillator is a fundamental example in physics where some nontrivial explicit calculations can be done.

Bibliography

[1] O. Babelon, L. Cantini, and B. Douçot. A semi-classical study of the Jaynes-Cummings model. J. Stat. Mech. Theory Exp., (7):P07011, 45, 2009.  MR 2539236
[2] L. Charles. Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators. Comm. Partial Differential Equations, 28(9-10), 2003.  MR 2001172 |  Zbl 1038.53086
[3] Y. Colin de Verdière. A semi-classical inverse problem II: reconstruction of the potential. oai:hal.archives-ouvertes.fr:hal-00251590_v1 .
[4] Y. Colin de Verdière. Spectre conjoint d’opérateurs pseudo-différentiels qui commutent II. Math. Z., 171:51–73, 1980.  MR 566483 |  Zbl 0478.35073
[5] Y. Colin de Verdière and V. Guillemin. Semi-classical inverse problem I: Taylor expansions. preprint, hal-00250568.
[6] R. Cushman and J. J. Duistermaat. The quantum spherical pendulum. Bull. Amer. Math. Soc. (N.S.), 19:475–479, 1988.  MR 956603 |  Zbl 0658.58039
[7] J.-P. Dufour, P. Molino, and A. Toulet. Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko. C. R. Acad. Sci. Paris Sér. I Math., 318:949–952, 1994.  MR 1278158 |  Zbl 0808.58025
[8] J. J. Duistermaat. On global action-angle variables. Comm. Pure Appl. Math., 33:687–706, 1980.  MR 596430 |  Zbl 0439.58014
[9] H. R. Dullin. Semi-global symplectic invariants of the spherical pendulum. preprint arXiv:1108.4962.  MR 3017036
[10] C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via riemannian orbifolds. Invent. Math., 110(1):1–22, 1992.  MR 1181812 |  Zbl 0778.58068
[11] V. Guillemin, T. Paul, and A. Uribe. “Bottom of the well” semi-classical trace invariants. Math. Res. Lett., 14(4):711–719, 2007.  MR 2335997 |  Zbl 1140.58007
[12] H. Hezari. Inverse spectral problems for Schrödinger operators. Comm. Math. Phys., 288(3):1061–1088, 2009.  MR 2504865 |  Zbl 1170.81042
[13] A. Iantchenko, J. Sjöstrand, and M. Zworski. Birkhoff normal forms in semi-classical inverse problems. Math. Res. Lett., 9(2-3):337–362, 2002.  MR 1909649 |  Zbl pre01804060
[14] M. Kac. Can one hear the shape of a drum ? The American Math. Monthly, 73(4):1–23, 1966.  MR 201237 |  Zbl 0139.05603
[15] J. Milnor. Eigenvalues of the laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA, 51:542, 1964.  MR 162204 |  Zbl 0124.31202
[16] Á. Pelayo and S. Vũ Ngọc. First steps in a symplectic and spectral theory of integrable systems. in preparation.
[17] Á. Pelayo and S. Vũ Ngọc. Semitoric integrable systems on symplectic 4-manifolds. Invent. Math., 177(3):571–597, 2009.  MR 2534101 |  Zbl 1215.53071
[18] Á. Pelayo and S. Vũ Ngọc. Hamiltonian dynamics and spectral theory for spin-oscillators. arXiv:1005.0439, to appear in Comm. Math. Phys., 2010.  MR 2864789 |  Zbl pre06005044
[19] Á. Pelayo and S. Vũ Ngọc. Constructing integrable systems of semitoric type. Acta Math., 206:93–125, 2011.  MR 2784664 |  Zbl 1225.53074
[20] Á. Pelayo and S. Vũ Ngọc. Symplectic theory of completely integrable hamiltonian systems. to appear in Bull. AMS., 2011.  MR 2801777 |  Zbl 1230.37075
[21] D.A. Sadovskií and B.I. Zhilinskií. Monodromy, diabolic points, and angular momentum coupling. Phys. Lett. A, 256(4):235–244, 1999.  MR 1689376 |  Zbl 0934.81005
[22] S. Vũ Ngọc. Symplectic inverse spectral theory for pseudodifferential operators. HAL preprint, June 2008. To appear in a Volume dedicated to Hans Duistermaat.  MR 2809478 |  Zbl pre06084418
[23] S. Vũ Ngọc. Quantum monodromy in integrable systems. Commun. Math. Phys., 203(2):465–479, 1999.  MR 1697606 |  Zbl 0981.35015
[24] S. Vũ Ngọc. Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type. Comm. Pure Appl. Math., 53(2):143–217, 2000.  MR 1721373 |  Zbl 1027.81012
[25] S. Vũ Ngọc. On semi-global invariants for focus-focus singularities. Topology, 42(2):365–380, 2003.  MR 1941440 |  Zbl 1012.37041
[26] S. Vũ Ngọc and Ch. Wacheux. Normal forms for hamiltonian systems near a focus-focus singularity. Preprint hal-00577205, 2010.
[27] S. Zelditch. The inverse spectral problem. In Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., IX, pages 401–467. Int. Press, Somerville, MA, 2004. With an appendix by Johannes Sjöstrand and Maciej Zworski.  MR 2195415 |  Zbl 1061.58029
[28] S. Zelditch. Inverse spectral problem for analytic domains. I: Balian-bloch trace formula. Commun. Math. Phys., 248(2):357–407, 2004.  MR 2073139 |  Zbl 1086.58016
[29] S. Zelditch. Inverse spectral problem for analytic domains. II. $\mathbb{Z}_2$-symmetric domains. Ann. of Math. (2), 170(1):205–269, 2009.  MR 2521115 |  Zbl 1196.58016
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