Isospectrality for quantum toric integrable systems

Charles, Laurent; Pelayo, Álvaro; Ngoc, San Vũ

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 5, pp. 815-849.

Nous donnons une description complète du spectre de tout système intégrable torique quantique, au moyen de l'analyse microlocale des opérateurs de Toeplitz. Ceci résout la question de l'isospectralité pour cette classe de systèmes intégrables : le spectre semi-classique d'un système intégrable quantique torique détermine le système intégrable classique sous-jacent à symplectomorphisme près. Nous donnons aussi une description complète de la théorie spectrale semi-classique des systèmes intégrables toriques quantiques. Ces questions sont classiques en théorie spectrale et remontent aux travaux fondateurs de Colin de Verdière, Guillemin et Sternberg parmi d'autres dans les années 70 et 80.

We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdière, Guillemin, Sternberg and others in the 1970s and 1980s.

Classification : 58J50, 81Q20, 81S10, 47B35, 37J15
Keywords: semi-classical analysis, symplectic geometry, operator spectrum, Toeplitz operators, toric symplectic manifold
Mot clés : analyse semi-classique, géométrie symplectique, spectre d'opérateurs, opérateurs de Toeplitz, variété symplectique torique

[1] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. | MR

[2] P. Bérard, Transplantation et isospectralité. I, Math. Ann. 292 (1992), 547-559. | MR

[3] J. Brüning & E. Heintze, Spektrale Starrheit gewisser Drehflächen, Math. Ann. 269 (1984), 95-101. | MR

[4] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), 167-192. | MR

[5] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent, Asymptotic Anal. 1 (1988), 227-261. | MR

[6] A.-M. Charbonnel & G. Popov, A semi-classical trace formula for several commuting operators, Comm. Partial Differential Equations 24 (1999), 283-323. | MR

[7] L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003), 1-28. | MR

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

[9] L. Charles, Symbolic calculus for Toeplitz operators with half-form, J. Symplectic Geom. 4 (2006), 171-198. | MR

[10] L. Charles, Toeplitz operators and Hamiltonian torus actions, J. Funct. Anal. 236 (2006), 299-350. | MR

[11] L. Charles, Semi-classical properties of geometric quantization with metaplectic correction, Comm. Math. Phys. 270 (2007), 445-480. | MR

[12] L. Charles, On the quantization of polygon spaces, Asian J. Math. 14 (2010), 109-152. | MR

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

[14] Y. Colin De Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable, Math. Z. 171 (1980), 51-73. | MR

[15] Y. Colin De Verdière, A semi-classical inverse problem II: reconstruction of the potential, in Geometric aspects of analysis and mechanics, Progr. Math. 292, Birkhäuser, 2011, 97-119. | MR

[16] Y. Colin De Verdière& V. Guillemin, A semi-classical inverse problem I: Taylor expansions, in Geometric aspects of analysis and mechanics, Progr. Math. 292, Birkhäuser, 2011, 81-95. | MR

[17] D. A. Cox, J. B. Little & H. K. Schenck, Toric varieties, Graduate Studies in Math. 124, Amer. Math. Soc., 2011. | MR

[18] C. B. Croke & V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology 37 (1998), 1265-1273. | MR

[19] V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), 85-134, 247. | MR

[20] K. Datchev, H. Hezari & I. Ventura, Spectral uniqueness of radial semiclassical Schrödinger operators, Math. Res. Lett. 18 (2011), 521-529. | MR

[21] T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. | MR

[22] E. B. Dryden, V. Guillemin & R. Sena-Dias, Hearing Delzant polytopes from the equivariant spectrum, to appear in Trans. Amer. Math. Soc. | MR

[23] J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207-281. | MR

[24] J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706. | MR

[25] J. J. Duistermaat, Principal fiber bundles, in Notes for the Utrecht Spring School, 2004, http://www.projects.science.uu.nl/Duistermaat/www/homepageHD/pf.pdf.

[26] J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the spin-$c$ Dirac operator, Modern Birkhäuser Classics, Birkhäuser, 2011. | MR

[27] J. J. Duistermaat & L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 183-269. | MR

[28] J. J. Duistermaat & Á. Pelayo, Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambridge Philos. Soc. 146 (2009), 695-718. | MR

[29] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verhandlungen der Deutschen Physikalischen Gesellschaft 19 (1917), 82-92.

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

[31] M. D. Garay & D. Van Straten, Classical and quantum integrability, Mosc. Math. J. 10 (2010), 519-545, 661. | MR

[32] C. Gordon, D. L. Webb & S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110 (1992), 1-22. | MR

[33] C. Gordon, D. L. Webb & S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 134-138. | MR

[34] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Math. 122, Birkhäuser, 1994. | MR

[35] V. Guillemin & D. Kazhdan, Some inverse spectral results for negatively curved $2$-manifolds, Topology 19 (1980), 301-312. | MR

[36] V. Guillemin & S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. | MR

[37] V. Guillemin & S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. | MR

[38] V. Guillemin & S. Sternberg, Symplectic techniques in physics, second éd., Cambridge Univ. Press, 1990. | MR

[39] L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79-183. | MR

[40] A. Iantchenko, J. Sjöstrand & M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett. 9 (2002), 337-362. | MR

[41] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23. | MR

[42] Y. Karshon, L. Kessler & M. Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom. 5 (2007), 139-166. | MR

[43] B. Kostant, Orbits, symplectic structures and representation theory, in Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), Nippon Hyoronsha, Tokyo, 1966, p. 71. | MR

[44] B. Kostant, Quantization and unitary representations. I. Prequantization, in Lectures in modern analysis and applications, III, Springer, 1970, 87-208. Lecture Notes in Math., Vol. 170. | MR

[45] B. Kostant, Symplectic spinors, in Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, 1974, 139-152. | MR

[46] B. Kostant, On the definition of quantization, in Géométrie symplectique et physique mathématique (Colloq. Internat. CNRS, No. 237, Aix-en-Provence, 1974), Éditions du CNRS, Paris, 1975, 187-210. | MR

[47] B. Kostant & Á. Pelayo, Geometric quantization, a Lie theory approach, to appear as Springer Universitext.

[48] R. De La Llave, A tutorial on KAM theory, in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math. 69, Amer. Math. Soc., 2001, 175-292. | MR

[49] I. Madsen & J. Tornehave, From calculus to cohomology, Cambridge Univ. Press, 1997. | MR

[50] H. P. J. Mckean & I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), 43-69. | MR

[51] A. Melin & J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérisque 284 (2003), 181-244. | MR

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

[53] L. Boutet De Monvel & V. Guillemin, The spectral theory of Toeplitz operators, Annals of Math. Studies 99, Princeton Univ. Press, 1981. | MR

[54] B. Osgood, R. Phillips & P. Sarnak, Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), 5359-5361. | MR

[55] B. Osgood, R. Phillips & P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), 212-234. | MR

[56] B. Osgood, R. Phillips & P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. 129 (1989), 293-362. | MR

[57] Á. Pelayo & S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math. 177 (2009), 571-597. | MR

[58] Á. Pelayo & S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math. 206 (2011), 93-125. | MR

[59] Á. Pelayo & S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.) 48 (2011), 409-455. | MR

[60] Á. Pelayo & S. Vũ Ngọc, First steps in symplectic and spectral theory of integrable systems, Discrete Contin. Dyn. Syst. 32 (2012), 3325-3377. | MR

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

[62] J.-M. Souriau, Quantification géométrique, Comm. Math. Phys. 1 (1966), 374-398. | MR

[63] J.-M. Souriau, Structure des systèmes dynamiques, Maîtrise de mathématiques, Dunod, 1970. | MR

[64] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169-186. | MR

[65] J. A. Toth, On the quantum expected values of integrable metric forms, J. Differential Geom. 52 (1999), 327-374. | MR

[66] J. A. Toth & S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111 (2002), 97-132. | MR

[67] J. A. Toth & S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré 4 (2003), 343-368. | MR

[68] G. M. Tuynman, Quantization: towards a comparison between methods, J. Math. Phys. 28 (1987), 2829-2840. | MR

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

[70] S. Vũ Ngọc, Systèmes intégrables semi-classiques : du local au global, Panoramas et Synthèses 22 (2006).

[71] S. Vũ Ngọc, Symplectic inverse spectral theory for pseudodifferential operators, in Geometric aspects of analysis and mechanics, Progr. Math. 292, Birkhäuser, 2011, 353-372. | MR

[72] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Göttinger Nachrichten (1911), 110-117.

[73] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), 441-479. | MR

[74] S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), 305-363. | MR

[75] S. Zelditch, Inverse spectral problem for analytic domains 170 (2009), 205-269. | MR