# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
##### References:
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Takens, Geometry of KAM tori for nearly integrable Hamiltonian systems,, Ergodic Theory Dynam. Systems, 27 (2007), 725. Google Scholar [8] J. Brüning and E. Heintze, Spektrale Starrheit gewisser Drehflächen,, Math. Ann., 269 (1984), 95. Google Scholar [9] P. Buser, Isospectral Riemann surfaces,, Ann. Inst. Fourier (Grenoble), 36 (1986), 167. Google Scholar [10] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudo-différentiels qui commutent,, Asymptotic Analysis, 1 (1988), 227. Google Scholar [11] A.-M. Charbonnel and G. Popov, A semi-classical trace formula for several commuting operators,, Comm. Partial Differential Equations, 24 (1999), 283. Google Scholar [12] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators,, Comm. Partial Differential Equations, 28 (2003), 1527. Google Scholar [13] L. Charles, Symbolic calculus for Toeplitz operators with half-forms,, Journal of Symplectic Geometry, 4 (2006), 171. Google Scholar [14] L. Charles, Á. Pelayo and S. Vũ Ngọc, Isospectrality for quantum toric integrable systems,, preprint, (). Google Scholar [15] L. Charles, Á. Pelayo and S. Vũ Ngọc, The inverse spectral conjecture for semitoric systems,, preprint., (). Google Scholar [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. Google Scholar [17] Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. I. Le cas non intégrable,, Duke Math. J., 46 (1979), 169. Google Scholar [18] Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II. Le cas intégrable,, Math. Z., 171 (1980), 51. Google Scholar [19] T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment,, Bull. Soc. Math. France, 116 (1988), 315. Google Scholar [20] J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfoldings of singularities,, Comm. Pure Appl. Math., 27 (1974), 207. Google Scholar [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. Google Scholar [22] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein,, Deutsche Physikalische Gesellschaft. Verhandlungen, 19 (1917), 82. Google Scholar [23] Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, \arXiv{1006.2501}., (). Google Scholar [24] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case,, Comment. Math. Helv., 65 (1990), 4. Google Scholar [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). Google Scholar [26] M. Garay, A rigidity theorem for Lagrangian deformations,, Compos. Math., 141 (2005), 1602. Google Scholar [27] M. Garay, Stable moment mappings and singular Lagrangian fibrations,, Q. J. Math., 56 (2005), 357. Google Scholar [28] C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds,, Invent. Math., 110 (1992), 1. Google Scholar [29] M. Gross, Topological mirror symmetry,, Invent. Math., 144 (2001), 75. Google Scholar [30] M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data. I,, J. Diff. Geom., 72 (2006), 169. Google Scholar [31] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping,, Invent. Math., 67 (1982), 491. Google Scholar [32] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations,, Invent. Math., 67 (1982), 515. Google Scholar [33] N. Hitchin, Stable bundles and integrable systems,, Duke Math. J., 54 (1987), 91. Google Scholar [34] M. Hitrik and J. Sjöstrand and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for nonselfadjoint operators,, Amer. J. Math., 129 (2007), 105. Google Scholar [35] M. Kac, Can one hear the shape of a drum?,, (Polish) Translated from the English (Amer. Math. Monthly, 73 (1966), 1. Google Scholar [36] A. Katok, Open problems in elliptic dynamics,, \url{http://www.math.psu.edu/katok_a/elliptic.pdf}, (). Google Scholar [37] F. Kirwan, Convexity properties of the moment mapping. III,, Invent. Math., 77 (1984), 547. Google Scholar [38] M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces,, in, 244 (2006), 321. Google Scholar [39] S. Kowalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. Google Scholar [40] P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467. Google Scholar [41] Y. Le Floch, Singular Bohr-Sommerfeld conditions in dimension 1: The elliptic case,, preprint, (2012). Google Scholar [42] N. C. Leung and M. Symington, Almost toric symplectic four-manifolds,, J. Symplectic Geom., 8 (2010), 143. Google Scholar [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. Google Scholar [44] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964). Google Scholar [45] B. Osgood, R. Phillips and P. Sarnak, Moduli space, heights and isospectral sets of plane domains,, Ann. of Math. (2), 129 (1989), 293. Google Scholar [46] Á. Pelayo, T. S. Ratiu and S. Vũ Ngọc, Symplectic bifurcation theory for integrable systems,, \arXiv{1108.0328}., (). Google Scholar [47] Á. Pelayo, V. Voevodsky and M. Warren, Basic $p$-adic analysis in the univalent foundations,, in preparation., (). Google Scholar [48] Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic $4$-manifolds,, Invent. Math., 177 (2009), 571. Google Scholar [49] Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type,, Acta Math., 206 (2011), 93. Google Scholar [50] Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems,, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409. Google Scholar [51] Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators,, Comm. Math Phys., 309 (2012), 123. Google Scholar [52] N. Reshetikhin, Lectures on the integrability of the six-vertex model,, in, (2010), 197. Google Scholar [53] D. A. Sadovskií and B. Zhilinskií, Counting levels within vibrational polyads,, J. Chem. Phys., 103 (1995). Google Scholar [54] C. Sevenheck and D. van Straten, Rigid and complete intersection Lagrangian singularities,, Manuscripta Math., 114 (2004), 197. Google Scholar [55] C. Sevenheck and D. van Straten, Deformation of singular Lagrangian subvarieties,, Math. Ann., 327 (2003), 79. Google Scholar [56] M. Symington, Four dimensions from two in symplectic topology,, in, 71 (2003), 153. Google Scholar [57] J. Toth and S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case,, Ann. Henri Poincaré, 4 (2003), 343. Google Scholar [58] V. Voevodsky, Univalent Foundations Project., Avaiable from: \url{http://www.math.ias.edu/~vladimir/.../univalent_foundations_project.pdf}., (). Google Scholar [59] S. Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type,, Comm. Pure Appl. Math., 53 (2000), 143. Google Scholar [60] S. Vũ Ngọc, Symplectic inverse spectral theory for pseudodifferential operators,, in, 292 (2011), 353. Google Scholar [61] S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy,, Adv. Math., 208 (2007), 909. Google Scholar [62] C. Wacheux, "About the Image of Semi-Toric Moment Map,", Ph.D thesis, (). Google Scholar [63] S. Zelditch, Inverse spectral problem for analytic domains. II. $\mathbbZ_2$-symmetric domains,, Ann. of Math. (2), 170 (2009), 205. Google Scholar [64] N. T. Zung, Convergence versus integrability in Birkhoff normal form,, Ann. of Math., 161 (2005), 141. Google Scholar [65] N. T. Zung, Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems,, Regul. Chaotic Dyn., 12 (2007), 680. Google Scholar

show all references

##### References:
 [1] V. I. Arnol'd, A theorem of Liouville concerning integrable problems of dynamics,, Sibirsk. Mat. Ž., 4 (1963), 471. Google Scholar [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 (1985). Google Scholar [3] M. Atiyah, Convexity and commuting Hamiltonians,, Bull. Lond. Math. Soc., 14 (1982), 1. Google Scholar [4] P. Bérard, Transplantation et isospectralité. I,, Math. Ann., 292 (1992), 547. Google Scholar [5] A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems. Geometry, Topology, Classification,", Translated from the 1999 Russian original, (1999). Google Scholar [6] B. Bramham and H. Hofer, First steps towards a symplectic dynamics,, to appear in Surveys in Differential Geometry (SDG), 17 (). Google Scholar [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. Google Scholar [8] J. Brüning and E. Heintze, Spektrale Starrheit gewisser Drehflächen,, Math. Ann., 269 (1984), 95. Google Scholar [9] P. Buser, Isospectral Riemann surfaces,, Ann. Inst. Fourier (Grenoble), 36 (1986), 167. Google Scholar [10] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudo-différentiels qui commutent,, Asymptotic Analysis, 1 (1988), 227. Google Scholar [11] A.-M. Charbonnel and G. Popov, A semi-classical trace formula for several commuting operators,, Comm. Partial Differential Equations, 24 (1999), 283. Google Scholar [12] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators,, Comm. Partial Differential Equations, 28 (2003), 1527. Google Scholar [13] L. Charles, Symbolic calculus for Toeplitz operators with half-forms,, Journal of Symplectic Geometry, 4 (2006), 171. Google Scholar [14] L. Charles, Á. Pelayo and S. Vũ Ngọc, Isospectrality for quantum toric integrable systems,, preprint, (). Google Scholar [15] L. Charles, Á. Pelayo and S. Vũ Ngọc, The inverse spectral conjecture for semitoric systems,, preprint., (). Google Scholar [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. Google Scholar [17] Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. I. Le cas non intégrable,, Duke Math. J., 46 (1979), 169. Google Scholar [18] Y. Colin de Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II. Le cas intégrable,, Math. Z., 171 (1980), 51. Google Scholar [19] T. Delzant, Hamiltoniens périodiques et image convexe de l'application moment,, Bull. Soc. Math. France, 116 (1988), 315. Google Scholar [20] J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfoldings of singularities,, Comm. Pure Appl. Math., 27 (1974), 207. Google Scholar [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. Google Scholar [22] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein,, Deutsche Physikalische Gesellschaft. Verhandlungen, 19 (1917), 82. Google Scholar [23] Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, \arXiv{1006.2501}., (). Google Scholar [24] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case,, Comment. Math. Helv., 65 (1990), 4. Google Scholar [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). Google Scholar [26] M. Garay, A rigidity theorem for Lagrangian deformations,, Compos. Math., 141 (2005), 1602. Google Scholar [27] M. Garay, Stable moment mappings and singular Lagrangian fibrations,, Q. J. Math., 56 (2005), 357. Google Scholar [28] C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds,, Invent. Math., 110 (1992), 1. Google Scholar [29] M. Gross, Topological mirror symmetry,, Invent. Math., 144 (2001), 75. Google Scholar [30] M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data. I,, J. Diff. Geom., 72 (2006), 169. Google Scholar [31] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping,, Invent. Math., 67 (1982), 491. Google Scholar [32] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations,, Invent. Math., 67 (1982), 515. Google Scholar [33] N. Hitchin, Stable bundles and integrable systems,, Duke Math. J., 54 (1987), 91. Google Scholar [34] M. Hitrik and J. Sjöstrand and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for nonselfadjoint operators,, Amer. J. Math., 129 (2007), 105. Google Scholar [35] M. Kac, Can one hear the shape of a drum?,, (Polish) Translated from the English (Amer. Math. Monthly, 73 (1966), 1. Google Scholar [36] A. Katok, Open problems in elliptic dynamics,, \url{http://www.math.psu.edu/katok_a/elliptic.pdf}, (). Google Scholar [37] F. Kirwan, Convexity properties of the moment mapping. III,, Invent. Math., 77 (1984), 547. Google Scholar [38] M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces,, in, 244 (2006), 321. Google Scholar [39] S. Kowalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. Google Scholar [40] P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467. Google Scholar [41] Y. Le Floch, Singular Bohr-Sommerfeld conditions in dimension 1: The elliptic case,, preprint, (2012). Google Scholar [42] N. C. Leung and M. Symington, Almost toric symplectic four-manifolds,, J. Symplectic Geom., 8 (2010), 143. Google Scholar [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. Google Scholar [44] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964). Google Scholar [45] B. Osgood, R. Phillips and P. Sarnak, Moduli space, heights and isospectral sets of plane domains,, Ann. of Math. (2), 129 (1989), 293. Google Scholar [46] Á. Pelayo, T. S. Ratiu and S. Vũ Ngọc, Symplectic bifurcation theory for integrable systems,, \arXiv{1108.0328}., (). Google Scholar [47] Á. Pelayo, V. Voevodsky and M. Warren, Basic $p$-adic analysis in the univalent foundations,, in preparation., (). Google Scholar [48] Á. Pelayo and S. Vũ Ngọc, Semitoric integrable systems on symplectic $4$-manifolds,, Invent. Math., 177 (2009), 571. Google Scholar [49] Á. Pelayo and S. Vũ Ngọc, Constructing integrable systems of semitoric type,, Acta Math., 206 (2011), 93. Google Scholar [50] Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems,, Bull. Amer. Math. Soc. (N.S.), 48 (2011), 409. Google Scholar [51] Á. Pelayo and S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators,, Comm. Math Phys., 309 (2012), 123. Google Scholar [52] N. Reshetikhin, Lectures on the integrability of the six-vertex model,, in, (2010), 197. Google Scholar [53] D. A. Sadovskií and B. Zhilinskií, Counting levels within vibrational polyads,, J. Chem. Phys., 103 (1995). Google Scholar [54] C. Sevenheck and D. van Straten, Rigid and complete intersection Lagrangian singularities,, Manuscripta Math., 114 (2004), 197. Google Scholar [55] C. Sevenheck and D. van Straten, Deformation of singular Lagrangian subvarieties,, Math. Ann., 327 (2003), 79. Google Scholar [56] M. Symington, Four dimensions from two in symplectic topology,, in, 71 (2003), 153. Google Scholar [57] J. Toth and S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case,, Ann. Henri Poincaré, 4 (2003), 343. Google Scholar [58] V. Voevodsky, Univalent Foundations Project., Avaiable from: \url{http://www.math.ias.edu/~vladimir/.../univalent_foundations_project.pdf}., (). Google Scholar [59] S. Vũ Ngọc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type,, Comm. Pure Appl. Math., 53 (2000), 143. Google Scholar [60] S. Vũ Ngọc, Symplectic inverse spectral theory for pseudodifferential operators,, in, 292 (2011), 353. Google Scholar [61] S. Vũ Ngọc, Moment polytopes for symplectic manifolds with monodromy,, Adv. Math., 208 (2007), 909. Google Scholar [62] C. Wacheux, "About the Image of Semi-Toric Moment Map,", Ph.D thesis, (). Google Scholar [63] S. Zelditch, Inverse spectral problem for analytic domains. II. $\mathbbZ_2$-symmetric domains,, Ann. of Math. (2), 170 (2009), 205. Google Scholar [64] N. T. Zung, Convergence versus integrability in Birkhoff normal form,, Ann. of Math., 161 (2005), 141. Google Scholar [65] N. T. Zung, Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems,, Regul. Chaotic Dyn., 12 (2007), 680. Google Scholar
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