August  2017, 37(8): 4461-4487. doi: 10.3934/dcds.2017191

Normalization in Banach scale Lie algebras via mould calculus and applications

1. 

CMLS, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France

2. 

CNRS UMR 8028 – IMCCE, Observatoire de Paris, 75014 Paris, France

Received  July 2016 Revised  March 2017 Published  April 2017

Fund Project: This work has been partially carried out thanks to the support of the A*MIDEX project (no ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR). D.S.'s work has received funding from the French National Research Agency under the reference ANR-12-BS01-0017.

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.

Citation: Thierry Paul, David Sauzin. Normalization in Banach scale Lie algebras via mould calculus and applications. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4461-4487. doi: 10.3934/dcds.2017191
References:
[1]

V. Arnol'd, Méthodes mathématiques de la mécanique classique, Mir, Moscou, 1976.

[2]

M. Bailey, Local classification of generalized complex structures, J. Differential Geom., 95 (2013), 1-37.  doi: 10.4310/jdg/1375124607.

[3]

D. BambusiS. Graffi and T. Paul, Normal forms and quantization formulae, Comm. Math. Phys., 207 (1999), 173-195.  doi: 10.1007/s002200050723.

[4]

G. D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. , 1966.

[5]

M. Born, Vorlesungen über Atommechanik, Springer, Berlin, (1925). English translation: The mechanics of the atom, Ungar, New-York, 1927.

[6]

L. Charles and S. Vũ Ngoc, Spectral asymptotics via the semiclassical Birkhoff normal form, Duke Math. J., 143 (2008), 463-511.  doi: 10.1215/00127094-2008-026.

[7]

M. Degli EspostiS. Graffi and J. Herczynski, Quantization of the classical Lie algorithm in the Bargmann representation, Annals of Physics, 209 (1991), 364-392.  doi: 10.1016/0003-4916(91)90034-6.

[8]

J. Écalle, Les Fonctions Résurgentes, Publ. Math. d'Orsay[Vol. 1: 81-05, Vol. 2: 81-06, Vol. 3: 85-05] 1981,1985.

[9]

J. Écalle, Six lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's conjecture, in Bifurcations and periodic orbits of vector fields (Montreal, PQ, 1992) (ed. by D. Schlomiuk), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , Kluwer Acad. Publ. , Dordrecht, 408 (1993), 75-184.

[10]

G. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, 1989. doi: 10.1515/9781400882427.

[11]

G. Gallavotti, The Elements of Mechanics, Springer Verlag, 1983. doi: 10.1007/978-3-662-00731-0.

[12]

S. Graffi and T. Paul, Schrödinger equation and canonical perturbation theory, Comm. Math. Phys., 108 (1987), 25-40.  doi: 10.1007/BF01210701.

[13]

S. Graffi and T. Paul, Convergence of a quantum normal form and an exact quantization formula, Journ. Func. Analysis, 262 (2012), 3340-3393.  doi: 10.1016/j.jfa.2012.01.010.

[14]

V. Guillemin and T. Paul, Some remarks about semiclassical trace invariants and quantum normal forms, Communication in Mathematical Physics, 294 (2010), 1-19.  doi: 10.1007/s00220-009-0920-3.

[15]

W. Heisenberg, Matrix mechanik, Zeitscrift für Physik., 33 (1925), 879-893. 

[16]

A. IantchenkoJ. Sjöstrand and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., 9 (2002), 337-362.  doi: 10.4310/MRL.2002.v9.n3.a9.

[17]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems, Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.

[18]

I. Marcut, Rigidity around Poisson submanifolds, Acta Math., 213 (2014), 137-198.  doi: 10.1007/s11511-014-0118-1.

[19]

E. MirandaP. Monnier and N. T. Zung, Rigidity of Hamiltonian actions on Poisson manifolds, Adv. Math., 229 (2012), 1136-1179.  doi: 10.1016/j.aim.2011.09.013.

[20]

P. Monnier and N. T. Zung, Levi decomposition for smooth Poisson structures, J. Differential Geom., 68 (2004), 347-395.  doi: 10.4310/jdg/1115669514.

[21]

J. K. Moser and C. L. Siegel, Lectures on celestial mechanics Classics in Mathematics. Springer Verlag, Berlin, 1995.

[22]

T. Paul and D. Sauzin, Normalization in Lie algebras via mould calculus and applications, preprint, hal-01298047.

[23]

T. Paul and L. Stolovitch, Quantum singular complete integrability, J. Funct. Analysis, 271 (2016), 1377-1433.  doi: 10.1016/j.jfa.2016.04.029.

[24]

R. Perez-Marco, Convergence or generic divergence of the Birkhoff normal form, Ann. of Math., 157 (2003), 557-574.  doi: 10.4007/annals.2003.157.557.

[25]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Volume 2, Gauthier-Villars, Paris, (1892), Blanchard, Paris, 1987.

[26]

J. Sjöstrand, Semi-excited levels in non-degenerate potential wells, Asymptotic analysis, 6 (1992), 29-43. 

[27]

L. Stolovitch, Progress in normal form theory, Nonlinearity, 22 (2009), R77-R99.  doi: 10.1088/0951-7715/22/7/R01.

[28]

H. Weyl, Group theory and quantum mechanics, (1928 in German), Dover Publications, NewYork, 1950.

[29]

N. T. Zung, Convergence versus integrability in normal form theory, Ann. of Math., 161 (2005), 141-156.  doi: 10.4007/annals.2005.161.141.

show all references

References:
[1]

V. Arnol'd, Méthodes mathématiques de la mécanique classique, Mir, Moscou, 1976.

[2]

M. Bailey, Local classification of generalized complex structures, J. Differential Geom., 95 (2013), 1-37.  doi: 10.4310/jdg/1375124607.

[3]

D. BambusiS. Graffi and T. Paul, Normal forms and quantization formulae, Comm. Math. Phys., 207 (1999), 173-195.  doi: 10.1007/s002200050723.

[4]

G. D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. Ⅸ American Mathematical Society, Providence, R. I. , 1966.

[5]

M. Born, Vorlesungen über Atommechanik, Springer, Berlin, (1925). English translation: The mechanics of the atom, Ungar, New-York, 1927.

[6]

L. Charles and S. Vũ Ngoc, Spectral asymptotics via the semiclassical Birkhoff normal form, Duke Math. J., 143 (2008), 463-511.  doi: 10.1215/00127094-2008-026.

[7]

M. Degli EspostiS. Graffi and J. Herczynski, Quantization of the classical Lie algorithm in the Bargmann representation, Annals of Physics, 209 (1991), 364-392.  doi: 10.1016/0003-4916(91)90034-6.

[8]

J. Écalle, Les Fonctions Résurgentes, Publ. Math. d'Orsay[Vol. 1: 81-05, Vol. 2: 81-06, Vol. 3: 85-05] 1981,1985.

[9]

J. Écalle, Six lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's conjecture, in Bifurcations and periodic orbits of vector fields (Montreal, PQ, 1992) (ed. by D. Schlomiuk), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , Kluwer Acad. Publ. , Dordrecht, 408 (1993), 75-184.

[10]

G. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, 1989. doi: 10.1515/9781400882427.

[11]

G. Gallavotti, The Elements of Mechanics, Springer Verlag, 1983. doi: 10.1007/978-3-662-00731-0.

[12]

S. Graffi and T. Paul, Schrödinger equation and canonical perturbation theory, Comm. Math. Phys., 108 (1987), 25-40.  doi: 10.1007/BF01210701.

[13]

S. Graffi and T. Paul, Convergence of a quantum normal form and an exact quantization formula, Journ. Func. Analysis, 262 (2012), 3340-3393.  doi: 10.1016/j.jfa.2012.01.010.

[14]

V. Guillemin and T. Paul, Some remarks about semiclassical trace invariants and quantum normal forms, Communication in Mathematical Physics, 294 (2010), 1-19.  doi: 10.1007/s00220-009-0920-3.

[15]

W. Heisenberg, Matrix mechanik, Zeitscrift für Physik., 33 (1925), 879-893. 

[16]

A. IantchenkoJ. Sjöstrand and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., 9 (2002), 337-362.  doi: 10.4310/MRL.2002.v9.n3.a9.

[17]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems, Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1044-3.

[18]

I. Marcut, Rigidity around Poisson submanifolds, Acta Math., 213 (2014), 137-198.  doi: 10.1007/s11511-014-0118-1.

[19]

E. MirandaP. Monnier and N. T. Zung, Rigidity of Hamiltonian actions on Poisson manifolds, Adv. Math., 229 (2012), 1136-1179.  doi: 10.1016/j.aim.2011.09.013.

[20]

P. Monnier and N. T. Zung, Levi decomposition for smooth Poisson structures, J. Differential Geom., 68 (2004), 347-395.  doi: 10.4310/jdg/1115669514.

[21]

J. K. Moser and C. L. Siegel, Lectures on celestial mechanics Classics in Mathematics. Springer Verlag, Berlin, 1995.

[22]

T. Paul and D. Sauzin, Normalization in Lie algebras via mould calculus and applications, preprint, hal-01298047.

[23]

T. Paul and L. Stolovitch, Quantum singular complete integrability, J. Funct. Analysis, 271 (2016), 1377-1433.  doi: 10.1016/j.jfa.2016.04.029.

[24]

R. Perez-Marco, Convergence or generic divergence of the Birkhoff normal form, Ann. of Math., 157 (2003), 557-574.  doi: 10.4007/annals.2003.157.557.

[25]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Volume 2, Gauthier-Villars, Paris, (1892), Blanchard, Paris, 1987.

[26]

J. Sjöstrand, Semi-excited levels in non-degenerate potential wells, Asymptotic analysis, 6 (1992), 29-43. 

[27]

L. Stolovitch, Progress in normal form theory, Nonlinearity, 22 (2009), R77-R99.  doi: 10.1088/0951-7715/22/7/R01.

[28]

H. Weyl, Group theory and quantum mechanics, (1928 in German), Dover Publications, NewYork, 1950.

[29]

N. T. Zung, Convergence versus integrability in normal form theory, Ann. of Math., 161 (2005), 141-156.  doi: 10.4007/annals.2005.161.141.

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