Isospectrality for quantum toric integrable systems

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1. [1] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc.14 (1982), 1–15. Zbl0482.58013MR642416
2. [2] P. Bérard, Transplantation et isospectralité. I, Math. Ann. 292 (1992), 547–559. MR1152950
3. [3] J. Brüning & E. Heintze, Spektrale Starrheit gewisser Drehflächen, Math. Ann.269 (1984), 95–101. Zbl0553.53028MR756778
4. [4] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble) 36 (1986), 167–192. Zbl0579.53036MR850750
5. [5] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d’opérateurs pseudodifférentiels qui commutent, Asymptotic Anal.1 (1988), 227–261. Zbl0665.35080MR962310
6. [6] A.-M. Charbonnel & G. Popov, A semi-classical trace formula for several commuting operators, Comm. Partial Differential Equations24 (1999), 283–323. Zbl0927.35138MR1672009
7. [7] L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys.239 (2003), 1–28. Zbl1059.47030MR1997113
8. [8] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Comm. Partial Differential Equations28 (2003), 1527–1566. Zbl1038.53086MR2001172
9. [9] L. Charles, Symbolic calculus for Toeplitz operators with half-form, J. Symplectic Geom.4 (2006), 171–198. Zbl1123.58016MR2275003
10. [10] L. Charles, Toeplitz operators and Hamiltonian torus actions, J. Funct. Anal.236 (2006), 299–350. Zbl1099.53059MR2227136
11. [11] L. Charles, Semi-classical properties of geometric quantization with metaplectic correction, Comm. Math. Phys.270 (2007), 445–480. Zbl1118.53059MR2276452
12. [12] L. Charles, On the quantization of polygon spaces, Asian J. Math.14 (2010), 109–152. Zbl1206.47095MR2726596
13. [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. Zbl0411.35073MR523605
14. [14] Y. Colin de Verdière, Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable, Math. Z. 171 (1980), 51–73. Zbl0478.35073MR566483
15. [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. MR2809469
16. [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. MR2809468
17. [17] D. A. Cox, J. B. Little & H. K. Schenck, Toric varieties, Graduate Studies in Math. 124, Amer. Math. Soc., 2011. MR2810322
18. [18] C. B. Croke & V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology37 (1998), 1265–1273. Zbl0936.58013MR1632920
19. [19] V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), 85–134, 247. Zbl0425.14013MR495499
20. [20] K. Datchev, H. Hezari & I. Ventura, Spectral uniqueness of radial semiclassical Schrödinger operators, Math. Res. Lett.18 (2011), 521–529. Zbl1241.35217MR2802585
21. [21] T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France116 (1988), 315–339. Zbl0676.58029MR984900
22. [22] E. B. Dryden, V. Guillemin & R. Sena-Dias, Hearing Delzant polytopes from the equivariant spectrum, to appear in Trans. Amer. Math. Soc. Zbl1244.58011MR2846357
23. [23] J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207–281. Zbl0285.35010MR405513
24. [24] J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math.33 (1980), 687–706. Zbl0439.58014MR596430
25. [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. [26] J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the spin-$c$ Dirac operator, Modern Birkhäuser Classics, Birkhäuser, 2011. Zbl1222.58014MR2809491
27. [27] J. J. Duistermaat & L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 183–269. Zbl0232.47055MR388464
28. [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. Zbl1163.53347MR2496353
29. [29] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verhandlungen der Deutschen Physikalischen Gesellschaft19 (1917), 82–92. 
30. [30] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv.65 (1990), 4–35. Zbl0702.58024MR1036125
31. [31] M. D. Garay & D. van Straten, Classical and quantum integrability, Mosc. Math. J. 10 (2010), 519–545, 661. Zbl1202.81098MR2732572
32. [32] C. Gordon, D. L. Webb & S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math.110 (1992), 1–22. Zbl0778.58068MR1181812
33. [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. Zbl0756.58049MR1136137
34. [34] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Math. 122, Birkhäuser, 1994. Zbl0828.58001MR1301331
35. [35] V. Guillemin & D. Kazhdan, Some inverse spectral results for negatively curved $2$-manifolds, Topology19 (1980), 301–312. Zbl0465.58027MR579579
36. [36] V. Guillemin & S. Sternberg, Convexity properties of the moment mapping, Invent. Math.67 (1982), 491–513. Zbl0503.58017MR664117
37. [37] V. Guillemin & S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math.67 (1982), 515–538. Zbl0503.58018MR664118
38. [38] V. Guillemin & S. Sternberg, Symplectic techniques in physics, second éd., Cambridge Univ. Press, 1990. Zbl0734.58005MR1066693
39. [39] L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183. Zbl0212.46601MR388463
40. [40] A. Iantchenko, J. Sjöstrand & M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett.9 (2002), 337–362. Zbl1258.35208MR1909649
41. [41] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1–23. Zbl0139.05603MR201237
42. [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. Zbl1136.53060MR2377250
43. [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. MR213476
44. [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. MR294568
45. [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. MR400304
46. [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. MR488137
47. [47] B. Kostant & Á. Pelayo, Geometric quantization, a Lie theory approach, to appear as Springer Universitext. 
48. [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. Zbl1055.37064MR1858536
49. [49] I. Madsen & J. Tornehave, From calculus to cohomology, Cambridge Univ. Press, 1997. Zbl0884.57001MR1454127
50. [50] H. P. J. McKean & I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry1 (1967), 43–69. Zbl0198.44301MR217739
51. [51] A. Melin & J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérisque284 (2003), 181–244. Zbl1061.35186MR2003421
52. [52] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. Zbl0124.31202MR162204
53. [53] L. Boutet de Monvel & V. Guillemin, The spectral theory of Toeplitz operators, Annals of Math. Studies 99, Princeton Univ. Press, 1981. Zbl0469.47021MR620794
54. [54] B. Osgood, R. Phillips & P. Sarnak, Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A.85 (1988), 5359–5361. Zbl0674.30021MR952815
55. [55] B. Osgood, R. Phillips & P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal.80 (1988), 212–234. Zbl0653.53021MR960229
56. [56] B. Osgood, R. Phillips & P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math.129 (1989), 293–362. Zbl0677.58045MR986795
57. [57] Á. Pelayo & S. Vũ Ngọc, Semitoric integrable systems on symplectic 4-manifolds, Invent. Math.177 (2009), 571–597. Zbl1215.53071MR2534101
58. [58] Á. Pelayo & S. Vũ Ngọc, Constructing integrable systems of semitoric type, Acta Math.206 (2011), 93–125. Zbl1225.53074MR2784664
59. [59] Á. Pelayo & S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N.S.) 48 (2011), 409–455. Zbl1230.37075MR2801777
60. [60] Á. Pelayo & S. Vũ Ngọc, First steps in symplectic and spectral theory of integrable systems, Discrete Contin. Dyn. Syst.32 (2012), 3325–3377. Zbl1257.37038MR2945820
61. [61] Á. Pelayo & S. Vũ Ngọc, Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys.309 (2012), 123–154. Zbl1263.70022MR2864789
62. [62] J.-M. Souriau, Quantification géométrique, Comm. Math. Phys.1 (1966), 374–398. Zbl1148.81307MR207332
63. [63] J.-M. Souriau, Structure des systèmes dynamiques, Maîtrise de mathématiques, Dunod, 1970. MR260238
64. [64] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math.121 (1985), 169–186. Zbl0585.58047MR782558
65. [65] J. A. Toth, On the quantum expected values of integrable metric forms, J. Differential Geom.52 (1999), 327–374. Zbl0992.53063MR1758299
66. [66] J. A. Toth & S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J.111 (2002), 97–132. Zbl1022.58013MR1876442
67. [67] J. A. Toth & S. Zelditch, $L^p$ norms of eigenfunctions in the completely integrable case, Ann. Henri Poincaré4 (2003), 343–368. Zbl1028.58028MR1985776
68. [68] G. M. Tuynman, Quantization: towards a comparison between methods, J. Math. Phys.28 (1987), 2829–2840. Zbl0639.58035MR917637
69. [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. Zbl1027.81012MR1721373
70. [70] S. Vũ Ngọc, Systèmes intégrables semi-classiques : du local au global, Panoramas et Synthèses 22 (2006). 
71. [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. Zbl1277.47065MR2809478
72. [72] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Göttinger Nachrichten (1911), 110–117. Zbl42.0432.03
73. [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. Zbl43.0436.01MR1511670
74. [74] S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), 305–363. Zbl0865.47018MR1437187
75. [75] S. Zelditch, Inverse spectral problem for analytic domains. II. ${\mathbb{Z}}_2$-symmetric domains, Ann. of Math. 170 (2009), 205–269. Zbl1196.58016MR2521115