October  2012, 32(10): 3325-3377. doi: 10.3934/dcds.2012.32.3325

First steps in symplectic and spectral theory of integrable systems

1. 

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States

2. 

Institut Universitaire de France, Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

Received  April 2011 Revised  February 2012 Published  May 2012

The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. While it is difficult to know what the best approach to such a large classification task would be, it is possible to single out some promising directions and preliminary problems. This paper discusses them and hints at a possible path, still loosely defined, to arrive at a classification. It mainly relies on recent progress concerning integrable systems with only non-hyperbolic and non-degenerate singularities.
    This work originated in an attempt to develop a theory aimed at answering some questions in quantum spectroscopy. Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. A main goal of this paper is to emphasize the symplectic issues that are relevant to quantum mechanical integrable systems, and to propose a strategy to solve them.
Citation: Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
References:
[1]

V. I. Arnol'd, A theorem of Liouville concerning integrable problems of dynamics, Sibirsk. Mat. Ž., 4 (1963), 471-474.

[2]

V. I. Arnol'd, S. M. Guseĭn-Zade and A. N. Varchenko, "Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts," Monographs in Mathematics, 82, Birkhäuser Boston, Inc., Boston, MA, 1985.

[3]

M. Atiyah, Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc., 14 (1982), 1-15.

[4]

P. Bérard, Transplantation et isospectralité. I, Math. Ann., 292 (1992), 547-559.

[5]

A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems. Geometry, Topology, Classification," Translated from the 1999 Russian original, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[6]

B. Bramham and H. Hofer, First steps towards a symplectic dynamics, to appear in Surveys in Differential Geometry (SDG), 17, arXiv:1102.3723.

[7]

H. Broer, R. Cushman, F. Francesco and F. Takens, Geometry of KAM tori for nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, 27 (2007), 725-741.

[8]

J. Brüning and E. Heintze, Spektrale Starrheit gewisser Drehflächen, Math. Ann., 269 (1984), 95-101.

[9]

P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), 36 (1986), 167-192.

[10]

A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudo-différentiels qui commutent, Asymptotic Analysis, 1 (1988), 227-261.

[11]

A.-M. Charbonnel and G. Popov, A semi-classical trace formula for several commuting operators, Comm. Partial Differential Equations, 24 (1999), 283-323.

[12]

L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations, 28 (2003), 1527-1566.

[13]

L. Charles, Symbolic calculus for Toeplitz operators with half-forms, Journal of Symplectic Geometry, 4 (2006), 171-198.

[14]

L. Charles, Á. Pelayo and S. Vũ Ngọc, Isospectrality for quantum toric integrable systems, preprint, arXiv:1111.5985.

[15]

L. Charles, Á. Pelayo and S. Vũ Ngọc, The inverse spectral conjecture for semitoric systems, preprint.

[16]

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

[17]

Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. I. Le cas non intégrable, Duke Math. J., 46 (1979), 169-182.

[18]

Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II. Le cas intégrable, Math. Z., 171 (1980), 51-73.

[19]

T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339.

[20]

J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfoldings of singularities, Comm. Pure Appl. Math., 27 (1974), 207-281.

[21]

J. J. Duistermaat and Á. Pelayo, Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambr. Phil. Soc., 146 (2009), 695-718.

[22]

A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Deutsche Physikalische Gesellschaft. Verhandlungen, 19 (1917), 82-92.

[23]

Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifoldsarXiv:1006.2501.

[24]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case, Comment. Math. Helv., 65 (1990), 4-35.

[25]

N. J. Fitch, C. A. Weidner, L. P. Parazzoli, H. R. Dullin and H. J. Lewandowski, Experimental demonstration of classical Hamiltonian monodromy in the 1 : 1 : 2 resonant elastic pendulum, Phys. Rev. Lett., (2009), 034301.

[26]

M. Garay, A rigidity theorem for Lagrangian deformations, Compos. Math., 141 (2005), 1602-1614.

[27]

M. Garay, Stable moment mappings and singular Lagrangian fibrations, Q. J. Math., 56 (2005), 357-366.

[28]

C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.

[29]

M. Gross, Topological mirror symmetry, Invent. Math., 144 (2001), 75-137.

[30]

M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Diff. Geom., 72 (2006), 169-338.

[31]

V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.

[32]

V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math., 67 (1982), 515-538.

[33]

N. Hitchin, Stable bundles and integrable systems, Duke Math. J., 54 (1987), 91-114.

[34]

M. Hitrik and J. Sjöstrand and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for nonselfadjoint operators, Amer. J. Math., 129 (2007), 105-182.

[35]

M. Kac, Can one hear the shape of a drum?, (Polish) Translated from the English (Amer. Math. Monthly, 73 (1966), part II, 1-23), Wiadom. Mat. (2), 13 (1971), 11-35.

[36]

A. Katok, Open problems in elliptic dynamicshttp://www.math.psu.edu/katok_a/elliptic.pdf

[37]

F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math., 77 (1984), 547-552.

[38]

M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, in "The Unity of Mathematics," Progr. Math., 244, Birkhäuser Boston, (2006), 321-385.

[39]

S. Kowalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.

[40]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.

[41]

Y. Le Floch, Singular Bohr-Sommerfeld conditions in dimension 1: The elliptic case, preprint, 2012.

[42]

N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187.

[43]

A. Melin and J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2. Autour de l'analyse microlocale, Astérisque, 284 (2003), 181-244.

[44]

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542.

[45]

B. Osgood, R. Phillips and P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2), 129 (1989), 293-362.

[46]

Á. Pelayo, T. S. Ratiu and S. Vũ Ngọc, Symplectic bifurcation theory for integrable systemsarXiv:1108.0328.

[47]

Á. Pelayo, V. Voevodsky and M. Warren, Basic $p$-adic analysis in the univalent foundations, in preparation.

[48]

Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic $4$-manifolds, Invent. Math., 177 (2009), 571-597.

[49]

Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125.

[50]

Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455.

[51]

Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math Phys., 309 (2012), 123-154.

[52]

N. Reshetikhin, Lectures on the integrability of the six-vertex model, in "Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing," Oxford Univ. Press, Oxford, (2010), 197-266.

[53]

D. A. Sadovskií and B. Zhilinskií, Counting levels within vibrational polyads, J. Chem. Phys., 103 (1995), 10520, 17 pp.

[54]

C. Sevenheck and D. van Straten, Rigid and complete intersection Lagrangian singularities, Manuscripta Math., 114 (2004), 197-209.

[55]

C. Sevenheck and D. van Straten, Deformation of singular Lagrangian subvarieties, Math. Ann., 327 (2003), 79-102.

[56]

M. Symington, Four dimensions from two in symplectic topology, in "Topology and Geometry of Manifolds" (Athens, GA, 2001), Proc. Symp. Pure Math., 71, Amer. Math. Soc., Providence, RI, (2003), 153-208.

[57]

J. Toth and S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré, 4 (2003), 343-368.

[58]

V. Voevodsky, Univalent Foundations Project. Avaiable from: http://www.math.ias.edu/~vladimir/.../univalent_foundations_project.pdf.

[59]

S. Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math., 53 (2000), 143-217.

[60]

S. Vũ Ngọc, Symplectic inverse spectral theory for pseudodifferential operators, in "Geometric Aspects of Analysis and Mechanics," Progress in Mathematics, 292, Birkhäuser/Springer, New York, (2011), 353-372.

[61]

S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934.

[62]

C. Wacheux, "About the Image of Semi-Toric Moment Map," Ph.D thesis, University of Rennes 1, in progress.

[63]

S. Zelditch, Inverse spectral problem for analytic domains. II. $\mathbbZ_2$-symmetric domains, Ann. of Math. (2), 170 (2009), 205-269.

[64]

N. T. Zung, Convergence versus integrability in Birkhoff normal form, Ann. of Math., 161 (2005), 141-156.

[65]

N. T. Zung, Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems, Regul. Chaotic Dyn., 12 (2007), 680-688.

show all references

References:
[1]

V. I. Arnol'd, A theorem of Liouville concerning integrable problems of dynamics, Sibirsk. Mat. Ž., 4 (1963), 471-474.

[2]

V. I. Arnol'd, S. M. Guseĭn-Zade and A. N. Varchenko, "Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts," Monographs in Mathematics, 82, Birkhäuser Boston, Inc., Boston, MA, 1985.

[3]

M. Atiyah, Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc., 14 (1982), 1-15.

[4]

P. Bérard, Transplantation et isospectralité. I, Math. Ann., 292 (1992), 547-559.

[5]

A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems. Geometry, Topology, Classification," Translated from the 1999 Russian original, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[6]

B. Bramham and H. Hofer, First steps towards a symplectic dynamics, to appear in Surveys in Differential Geometry (SDG), 17, arXiv:1102.3723.

[7]

H. Broer, R. Cushman, F. Francesco and F. Takens, Geometry of KAM tori for nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems, 27 (2007), 725-741.

[8]

J. Brüning and E. Heintze, Spektrale Starrheit gewisser Drehflächen, Math. Ann., 269 (1984), 95-101.

[9]

P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), 36 (1986), 167-192.

[10]

A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudo-différentiels qui commutent, Asymptotic Analysis, 1 (1988), 227-261.

[11]

A.-M. Charbonnel and G. Popov, A semi-classical trace formula for several commuting operators, Comm. Partial Differential Equations, 24 (1999), 283-323.

[12]

L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations, 28 (2003), 1527-1566.

[13]

L. Charles, Symbolic calculus for Toeplitz operators with half-forms, Journal of Symplectic Geometry, 4 (2006), 171-198.

[14]

L. Charles, Á. Pelayo and S. Vũ Ngọc, Isospectrality for quantum toric integrable systems, preprint, arXiv:1111.5985.

[15]

L. Charles, Á. Pelayo and S. Vũ Ngọc, The inverse spectral conjecture for semitoric systems, preprint.

[16]

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

[17]

Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. I. Le cas non intégrable, Duke Math. J., 46 (1979), 169-182.

[18]

Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II. Le cas intégrable, Math. Z., 171 (1980), 51-73.

[19]

T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339.

[20]

J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfoldings of singularities, Comm. Pure Appl. Math., 27 (1974), 207-281.

[21]

J. J. Duistermaat and Á. Pelayo, Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambr. Phil. Soc., 146 (2009), 695-718.

[22]

A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Deutsche Physikalische Gesellschaft. Verhandlungen, 19 (1917), 82-92.

[23]

Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifoldsarXiv:1006.2501.

[24]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case, Comment. Math. Helv., 65 (1990), 4-35.

[25]

N. J. Fitch, C. A. Weidner, L. P. Parazzoli, H. R. Dullin and H. J. Lewandowski, Experimental demonstration of classical Hamiltonian monodromy in the 1 : 1 : 2 resonant elastic pendulum, Phys. Rev. Lett., (2009), 034301.

[26]

M. Garay, A rigidity theorem for Lagrangian deformations, Compos. Math., 141 (2005), 1602-1614.

[27]

M. Garay, Stable moment mappings and singular Lagrangian fibrations, Q. J. Math., 56 (2005), 357-366.

[28]

C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22.

[29]

M. Gross, Topological mirror symmetry, Invent. Math., 144 (2001), 75-137.

[30]

M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Diff. Geom., 72 (2006), 169-338.

[31]

V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.

[32]

V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math., 67 (1982), 515-538.

[33]

N. Hitchin, Stable bundles and integrable systems, Duke Math. J., 54 (1987), 91-114.

[34]

M. Hitrik and J. Sjöstrand and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for nonselfadjoint operators, Amer. J. Math., 129 (2007), 105-182.

[35]

M. Kac, Can one hear the shape of a drum?, (Polish) Translated from the English (Amer. Math. Monthly, 73 (1966), part II, 1-23), Wiadom. Mat. (2), 13 (1971), 11-35.

[36]

A. Katok, Open problems in elliptic dynamicshttp://www.math.psu.edu/katok_a/elliptic.pdf

[37]

F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math., 77 (1984), 547-552.

[38]

M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, in "The Unity of Mathematics," Progr. Math., 244, Birkhäuser Boston, (2006), 321-385.

[39]

S. Kowalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.

[40]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.

[41]

Y. Le Floch, Singular Bohr-Sommerfeld conditions in dimension 1: The elliptic case, preprint, 2012.

[42]

N. C. Leung and M. Symington, Almost toric symplectic four-manifolds, J. Symplectic Geom., 8 (2010), 143-187.

[43]

A. Melin and J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2. Autour de l'analyse microlocale, Astérisque, 284 (2003), 181-244.

[44]

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542.

[45]

B. Osgood, R. Phillips and P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2), 129 (1989), 293-362.

[46]

Á. Pelayo, T. S. Ratiu and S. Vũ Ngọc, Symplectic bifurcation theory for integrable systemsarXiv:1108.0328.

[47]

Á. Pelayo, V. Voevodsky and M. Warren, Basic $p$-adic analysis in the univalent foundations, in preparation.

[48]

Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic $4$-manifolds, Invent. Math., 177 (2009), 571-597.

[49]

Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125.

[50]

Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409-455.

[51]

Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math Phys., 309 (2012), 123-154.

[52]

N. Reshetikhin, Lectures on the integrability of the six-vertex model, in "Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing," Oxford Univ. Press, Oxford, (2010), 197-266.

[53]

D. A. Sadovskií and B. Zhilinskií, Counting levels within vibrational polyads, J. Chem. Phys., 103 (1995), 10520, 17 pp.

[54]

C. Sevenheck and D. van Straten, Rigid and complete intersection Lagrangian singularities, Manuscripta Math., 114 (2004), 197-209.

[55]

C. Sevenheck and D. van Straten, Deformation of singular Lagrangian subvarieties, Math. Ann., 327 (2003), 79-102.

[56]

M. Symington, Four dimensions from two in symplectic topology, in "Topology and Geometry of Manifolds" (Athens, GA, 2001), Proc. Symp. Pure Math., 71, Amer. Math. Soc., Providence, RI, (2003), 153-208.

[57]

J. Toth and S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré, 4 (2003), 343-368.

[58]

V. Voevodsky, Univalent Foundations Project. Avaiable from: http://www.math.ias.edu/~vladimir/.../univalent_foundations_project.pdf.

[59]

S. Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math., 53 (2000), 143-217.

[60]

S. Vũ Ngọc, Symplectic inverse spectral theory for pseudodifferential operators, in "Geometric Aspects of Analysis and Mechanics," Progress in Mathematics, 292, Birkhäuser/Springer, New York, (2011), 353-372.

[61]

S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934.

[62]

C. Wacheux, "About the Image of Semi-Toric Moment Map," Ph.D thesis, University of Rennes 1, in progress.

[63]

S. Zelditch, Inverse spectral problem for analytic domains. II. $\mathbbZ_2$-symmetric domains, Ann. of Math. (2), 170 (2009), 205-269.

[64]

N. T. Zung, Convergence versus integrability in Birkhoff normal form, Ann. of Math., 161 (2005), 141-156.

[65]

N. T. Zung, Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems, Regul. Chaotic Dyn., 12 (2007), 680-688.

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