Comptes Rendus
no. 11
Fractional monodromy of resonant classical and quantum oscillators
[Monodromie fractionnelle des oscillateurs résonnants classiques et quantiques]
Nikolaı́ N. Nekhoroshev1, 2 ; Dmitriı́ A. Sadovskiı́2 ; Boris I. Zhilinskii2
1 Department of mathematics and mechanics, Moscow State University, Moscow, 119 899 Russia
2 Université du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France
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).

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Révisé le :
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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 
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     title = {Fractional monodromy of resonant classical and quantum oscillators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {985--988},
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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/

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