Spectral invariants for coupled spin-oscillators

San Vũ Ngọc[1]

  • [1] IRMAR (UMR 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-18
  • ISSN: 2266-0607

Abstract

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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.

How to cite

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Vũ Ngọc, San. "Spectral invariants for coupled spin-oscillators." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-18. <http://eudml.org/doc/251152>.

@article{VũNgọc2011-2012,
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.},
affiliation = {IRMAR (UMR 6625) Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France},
author = {Vũ Ngọc, San},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {inverse spectral theory; Schrödinger operator; Morse function; integrable systems; spin-oscillator coupling},
language = {eng},
pages = {1-18},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Spectral invariants for coupled spin-oscillators},
url = {http://eudml.org/doc/251152},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Vũ Ngọc, San
TI - Spectral invariants for coupled spin-oscillators
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 18
AB - 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.
LA - eng
KW - inverse spectral theory; Schrödinger operator; Morse function; integrable systems; spin-oscillator coupling
UR - http://eudml.org/doc/251152
ER -

References

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  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. MR2539236
  2. L. Charles. Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators. Comm. Partial Differential Equations, 28(9-10), 2003. Zbl1038.53086MR2001172
  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. Zbl0478.35073MR566483
  5. Y. Colin de Verdière and V. Guillemin. Semi-classical inverse problem I: Taylor expansions. preprint, hal-00250568. Zbl1258.34036
  6. R. Cushman and J. J. Duistermaat. The quantum spherical pendulum. Bull. Amer. Math. Soc. (N.S.), 19:475–479, 1988. Zbl0658.58039MR956603
  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. Zbl0808.58025MR1278158
  8. J. J. Duistermaat. On global action-angle variables. Comm. Pure Appl. Math., 33:687–706, 1980. Zbl0439.58014MR596430
  9. H. R. Dullin. Semi-global symplectic invariants of the spherical pendulum. preprint arXiv:1108.4962. Zbl1266.37024MR3017036
  10. C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via riemannian orbifolds. Invent. Math., 110(1):1–22, 1992. Zbl0778.58068MR1181812
  11. V. Guillemin, T. Paul, and A. Uribe. “Bottom of the well” semi-classical trace invariants. Math. Res. Lett., 14(4):711–719, 2007. Zbl1140.58007MR2335997
  12. H. Hezari. Inverse spectral problems for Schrödinger operators. Comm. Math. Phys., 288(3):1061–1088, 2009. Zbl1170.81042MR2504865
  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. Zbl1258.35208MR1909649
  14. M. Kac. Can one hear the shape of a drum ? The American Math. Monthly, 73(4):1–23, 1966. Zbl0139.05603MR201237
  15. J. Milnor. Eigenvalues of the laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA, 51:542, 1964. Zbl0124.31202MR162204
  16. Á. Pelayo and S. Vũ Ngọc. First steps in a symplectic and spectral theory of integrable systems. in preparation. Zbl1257.37038
  17. Á. Pelayo and S. Vũ Ngọc. Semitoric integrable systems on symplectic 4-manifolds. Invent. Math., 177(3):571–597, 2009. Zbl1215.53071MR2534101
  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. Zbl1263.70022MR2864789
  19. Á. Pelayo and S. Vũ Ngọc. Constructing integrable systems of semitoric type. Acta Math., 206:93–125, 2011. Zbl1225.53074MR2784664
  20. Á. Pelayo and S. Vũ Ngọc. Symplectic theory of completely integrable hamiltonian systems. to appear in Bull. AMS., 2011. Zbl1230.37075MR2801777
  21. D.A. Sadovskií and B.I. Zhilinskií. Monodromy, diabolic points, and angular momentum coupling. Phys. Lett. A, 256(4):235–244, 1999. Zbl0934.81005MR1689376
  22. S. Vũ Ngọc. Symplectic inverse spectral theory for pseudodifferential operators. HAL preprint, June 2008. To appear in a Volume dedicated to Hans Duistermaat. Zbl1277.47065MR2809478
  23. S. Vũ Ngọc. Quantum monodromy in integrable systems. Commun. Math. Phys., 203(2):465–479, 1999. Zbl0981.35015MR1697606
  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. Zbl1027.81012MR1721373
  25. S. Vũ Ngọc. On semi-global invariants for focus-focus singularities. Topology, 42(2):365–380, 2003. Zbl1012.37041MR1941440
  26. S. Vũ Ngọc and Ch. Wacheux. Normal forms for hamiltonian systems near a focus-focus singularity. Preprint hal-00577205, 2010. Zbl1303.37018
  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. Zbl1061.58029MR2195415
  28. S. Zelditch. Inverse spectral problem for analytic domains. I: Balian-bloch trace formula. Commun. Math. Phys., 248(2):357–407, 2004. Zbl1086.58016MR2073139
  29. S. Zelditch. Inverse spectral problem for analytic domains. II. ℤ 2 -symmetric domains. Ann. of Math. (2), 170(1):205–269, 2009. Zbl1196.58016MR2521115

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