# KAM Tori and Quantum Birkhoff Normal Forms

Georgi Popov^{[1]}

- [1] Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France

Séminaire Équations aux dérivées partielles (1999-2000)

- Volume: 1999-2000, page 1-13

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topPopov, Georgi. "KAM Tori and Quantum Birkhoff Normal Forms." Séminaire Équations aux dérivées partielles 1999-2000 (1999-2000): 1-13. <http://eudml.org/doc/10993>.

@article{Popov1999-2000,

abstract = {This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian $H$ close to a completely integrable one and a suitable Cantor set $\Theta $ defined by a Diophantine condition, we find a family $\Lambda _\omega ,\ \omega \in \Theta $, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta $ which is Gevrey smooth with respect to $\omega $ in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda $ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of $H$ around $\Lambda $. This leads to effective stability of the quasiperiodic motion near $\Lambda $. We investigate the semi-classical asymptotics of a Schrödinger type operator with a principal symbol $H$. We obtain semiclassical quasimodes with exponentially small error terms which are associated with the Gevrey family of KAM tori $\Lambda _\omega ,\ \omega \in \Theta $. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around $\Lambda $ in suitable Gevrey classes starting from the BNF of $H$. As an application, we obtain a sharp lower bound for the counting function of the resonances which are exponentially close to a suitable compact subinterval of the real axis.},

affiliation = {Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France},

author = {Popov, Georgi},

journal = {Séminaire Équations aux dérivées partielles},

keywords = {Kolmogorov-Arnold-Moser theorem; Gevrey classes; analytic Hamiltonians; effective stability; semiclassical asymptotics; Schrödinger operators; quasiperiodic motion },

language = {eng},

pages = {1-13},

publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {KAM Tori and Quantum Birkhoff Normal Forms},

url = {http://eudml.org/doc/10993},

volume = {1999-2000},

year = {1999-2000},

}

TY - JOUR

AU - Popov, Georgi

TI - KAM Tori and Quantum Birkhoff Normal Forms

JO - Séminaire Équations aux dérivées partielles

PY - 1999-2000

PB - Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 1999-2000

SP - 1

EP - 13

AB - This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian $H$ close to a completely integrable one and a suitable Cantor set $\Theta $ defined by a Diophantine condition, we find a family $\Lambda _\omega ,\ \omega \in \Theta $, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta $ which is Gevrey smooth with respect to $\omega $ in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda $ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of $H$ around $\Lambda $. This leads to effective stability of the quasiperiodic motion near $\Lambda $. We investigate the semi-classical asymptotics of a Schrödinger type operator with a principal symbol $H$. We obtain semiclassical quasimodes with exponentially small error terms which are associated with the Gevrey family of KAM tori $\Lambda _\omega ,\ \omega \in \Theta $. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around $\Lambda $ in suitable Gevrey classes starting from the BNF of $H$. As an application, we obtain a sharp lower bound for the counting function of the resonances which are exponentially close to a suitable compact subinterval of the real axis.

LA - eng

KW - Kolmogorov-Arnold-Moser theorem; Gevrey classes; analytic Hamiltonians; effective stability; semiclassical asymptotics; Schrödinger operators; quasiperiodic motion

UR - http://eudml.org/doc/10993

ER -

## References

top- N. Burq, Absence de résonance près du réel pour l’opérateur de Schrödinger, Seminair de l’Equations aux Dérivées Partielles, ${n}^{o}$ 17, Ecole Polytechnique, 1997/1998 Zbl1255.35085
- F. Cardoso, G. Popov, Quasimodes with exponentially small errors associated with broken elliptic rays, in preparation Zbl1136.35426
- Y. Colin de Verdière, Quasimodes sur les variétés Riemanniennes, Inventiones Math., Vol. 43, 1977, pp. 15-52 Zbl0449.53040MR501196
- V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993. Zbl0814.58001MR1239173
- P. Lochak, Canonical perturbation theory:an approach based on joint approximations, Uspekhi Mat. Nauk, Vol. 47, 6, 1992, pp. 59-140 (in Russian); translation in: Russian Math. Surveys, Vol. 47, 6, 1992, pp. 57-133. Zbl0795.58042MR1209145
- G. Popov, Invariant tori effective stability and quasimodes with exponentially small error term I - Birkhoff normal forms, Ann. Henri Poincaré, 2000, to appear. Zbl0970.37050MR1770799
- G. Popov, Invariant tori effective stability and quasimodes with exponentially small error term II - Quantum Birkhoff normal forms, Ann. Henri Poincaré, 2000, to appear. Zbl1002.37028MR1770800
- J. Pöschel, Lecture on the classical KAM Theorem, School on dynamical systems, May 1992, International center for science and high technology, Trieste, Italy
- J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., Vol. 213, 1993, pp. 187-217. Zbl0857.70009MR1221713
- J. Sjöstrand, A trace formula and review of some estimates for resonances. In: L. Rodino (eds.) Microlocal analysis and spectral theory. Nato ASI Series C: Mathematical and Physical Sciences, 490, pp. 377-437: Kluwer Academic Publishers 1997 Zbl0877.35090MR1451399
- J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, Journal of AMS, Vol. 4(4), 1991, pp. 729-769. Zbl0752.35046MR1115789
- Stefanov P.: Quasimodes and resonances: Sharp lower bounds, Duke Math. J., 99, 1, 1999, pp. 75-92. Zbl0952.47013MR1700740
- S.-H. Tang and M. Zworski, >From quasimodes to resonances, Math. Res. Lett., 5, 1998, pp. 261-272. Zbl0913.35101MR1637824

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