Comptes Rendus
Fractional monodromy of resonant classical and quantum oscillators
[Monodromie fractionnelle des oscillateurs résonnants classiques et quantiques]
Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 985-988.

La monodromie fractionnelle est introduite pour une classe de fibrations intégrables, qui apparaissent naturellement en mécanique classique dans le cas d'un oscillateur nonlinéaire avec résonance. On démontre, que la même monodromie fractionnelle caractérise de façon qualitative le réseau des états quantiques dans le spectre conjoint des observables pour les systèmes quantiques correspondants. Les résultats sont présentés en utilisant l'exemple d'un oscillateur à deux degrés de liberté avec la résonance 1 :(−1) et 1 :(−2).

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(−1) and 1:(−2).

Reçu le :
Révisé le :
Publié le :
DOI : 10.1016/S1631-073X(02)02584-0

Nikolaı́ N. Nekhoroshev 1, 2 ; Dmitriı́ A. Sadovskiı́ 2 ; Boris I. Zhilinskii 2

1 Department of mathematics and mechanics, Moscow State University, Moscow, 119 899 Russia
2 Université du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France
@article{CRMATH_2002__335_11_985_0,
     author = {Nikola{\i}́ N. Nekhoroshev and Dmitri{\i}́ A. Sadovski{\i}́ and Boris I. Zhilinskii},
     title = {Fractional monodromy of resonant classical and quantum oscillators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {985--988},
     publisher = {Elsevier},
     volume = {335},
     number = {11},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02584-0},
     language = {en},
}
TY  - JOUR
AU  - Nikolaı́ N. Nekhoroshev
AU  - Dmitriı́ A. Sadovskiı́
AU  - Boris I. Zhilinskii
TI  - Fractional monodromy of resonant classical and quantum oscillators
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 985
EP  - 988
VL  - 335
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02584-0
LA  - en
ID  - CRMATH_2002__335_11_985_0
ER  - 
%0 Journal Article
%A Nikolaı́ N. Nekhoroshev
%A Dmitriı́ A. Sadovskiı́
%A Boris I. Zhilinskii
%T Fractional monodromy of resonant classical and quantum oscillators
%J Comptes Rendus. Mathématique
%D 2002
%P 985-988
%V 335
%N 11
%I Elsevier
%R 10.1016/S1631-073X(02)02584-0
%G en
%F CRMATH_2002__335_11_985_0
Nikolaı́ N. Nekhoroshev; Dmitriı́ A. Sadovskiı́; Boris I. Zhilinskii. Fractional monodromy of resonant classical and quantum oscillators. Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 985-988. doi : 10.1016/S1631-073X(02)02584-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02584-0/

[1] V.I. Arnol'd Mathematical Methods of Classical Mechanics, Springer, New York, 1981

[2] A.V. Bolsinov; A.T. Fomenko Geometry and Topology of Integrable Geodesic Flows on Surfaces, Ser. Regular and Chaotic Dynamics, II, Editorial URSS, Moscow, 1999

[3] M.S. Child; T. Weston; J. Tennyson Quantum monodromy in the spectrum of H2O and other systems: New insight into the level structure of quasi linear molecules, Molecular Phys., Volume 96 (1999), pp. 371-379

[4] Y. Colin de Verdière, S. Vũ Ngo ˙c, Singular Bohr–Sommerfeld rules for 2D integrable systems, Preprint 508, Institut Fourier, 2000. Ann. Sci. École Norm. Sup., in press

[5] R.H. Cushman; L.M. Bates Global Aspects of Classical Integrable Systems, Birkhäuser, Basel, 1997

[6] R.H. Cushman; J.J. Duistermaat The quantum mechanical spherical pendulum, Bull. Am. Math. Soc., Volume 19 (1988), pp. 475-479

[7] R.H. Cushman; D.A. Sadovskiı́ Monodromy in the hydrogen atom, Physica D, Volume 65 (2000), pp. 166-196

[8] P. Dazord; T. Delzant Le problème général des variables action-angle, J. Differential Geom., Volume 26 (1987), pp. 223-251

[9] J.J. Duistermaat On global action angle coordinates, Comm. Pure Appl. Math., Volume 33 (1980), pp. 687-706

[10] F. Faure; B.I. Zhilinskii Topological Chern indexes in molecular spectra, Phys. Rev. Lett., Volume 85 (2000), pp. 960-963

[11] L. Grondin; D.A. Sadovskiı́; B.I. Zhilinskii Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra, Phys. Rev. A, Volume 142 (2002) (012105-1–15)

[12] L.M. Lerman; Ya.L. Umanskiı́ Four Dimensional Integrable Hamiltonian Systems with Simple Singular Points, Transl. Math. Monographs, 176, American Mathematical Society, Providence, RI, 1998

[13] V. Matveev Integrable Hamiltonian systems with two degrees of freedom. The topological structure of saturated neighborhoods of points of focus–focus and saddle–saddle type, Sb. Math., Volume 187 (1996) no. 4, pp. 495-524

[14] N.N. Nekhoroshev Action-angle variables and their generalizations, Trans. Moscow Math. Soc., Volume 26 (1972), pp. 180-198

[15] S. Vũ Ng Quantum monodromy in integrable systems, Comm. Math. Phys., Volume 203 (1999), pp. 465-479

[16] D.A. Sadovskiı́; B.I. Zhilinskii Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, Volume 256 (1999), pp. 235-244

[17] H. Waalkens; H.R. Dullin Quantum monodromy in ellipsoidal billiards, Ann. Phys. (N.Y.), Volume 295 (2001), pp. 81-112

[18] N.T. Zung A note on focus-focus singularities, Differential Geom. Appl., Volume 7 (1997), pp. 123-130 (Another note on focus–focus singularities Lett. Math. Phys., 60, 2002, pp. 87-89)

Cité par Sources :

Commentaires - Politique