Analytic Geometry and Semi-classical Analysis

M. D. Garay

Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 39-63.

Résumé

We study deformation theory for quantum integrable systems and prove several theorems concerning the Gevrey convergence and the unicity of perturbative expansions.

Bibliographie

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[2] Arnold V. I., “Kriticheskie tochki funktsii na mnogoobrazii s kraem, prostye gruppy Li $B_k$, $C_k$, $F_4$ i osobennosti evolyut”, UMN, 33:5 (1978), 91–105 | MR

[3] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii: Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR

[4] Birkhoff G. D., Dynamical systems, AMS Colloq. Publ., 9, Amer. Math. Soc., Providence, RI, 1966 | MR

[5] Boutet de Monvel L., Krée P., “Pseudo-differential operators and Gevrey classes”, Ann. Inst. Fourier, 17:1 (1967), 295–323 | MR | Zbl

[6] Brieskorn E., “Die Monodromie der isolierten Singularitäten von Hyperflächen”, Manuscr. math., 2 (1970), 103–161 | DOI | MR | Zbl

[7] Buchweitz R.-O., Greuel G.-M., “The Milnor number and deformations of complex curve singularities”, Invent. math., 58 (1980), 241–281 | DOI | MR | Zbl

[8] Cartan H., “Faisceaux analytiques sur les variétés de Stein: démonstration des théorèmes fondamentaux”, Sém. H. Cartan, 4:19 (1951–1952), 1–15 | MR

[9] Cartan H., Serre J.-P., “Un théorème de finitude concernant les variétés analytiques compactes”, C. r. Acad. sci. Paris, 237 (1953), 128–130 | MR | Zbl

[10] Colin de Verdière Y., “Singular Lagrangian manifolds and semiclassical analysis”, Duke Math. J., 116:2 (2003), 263–298 | DOI | MR | Zbl

[11] Colin de Verdière`Y., Parisse B., “Singular Bohr–Sommerfeld rules”, Commun. Math. Phys., 205 (1999), 459–500 | DOI | MR | Zbl

[12] Colin de Verdière Y., Vey J., “Le lemme de Morse isochore”, Topology, 18 (1979), 283–293 | DOI | MR | Zbl

[13] Dimassi M., Sjöstrand J., Spectral asymptotics in the semi-classical limit, LMS Lect. Note Ser., 268, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[14] Douady A., “Le théorème des images directes de Grauert (d'après Kiehl–Verdier)”, Astérisque, 16 (1974), 49–62 | MR | Zbl

[15] Du'c N. H., Pham F., “Germes de configurations legendriennes stables et fonctions d'Airy–Weber généralisées”, Ann. Inst. Fourier, 41:4 (1991), 905–936 | MR

[16] Eliasson L. H., “Normal forms for Hamiltonian systems with Poisson commuting integrals – elliptic case”, Comment. Math. Helv., 65:1 (1990), 4–35 | DOI | MR | Zbl

[17] Forster O., Knorr K., “Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange”, Manuscr. math., 5 (1971), 19–44 | DOI | MR | Zbl

[18] Garay M. D., “An isochore versal deformation theorem”, Topology, 43:5 (2004), 1081–1088 | DOI | MR | Zbl

[19] Garay M. D., “A rigidity theorem for Lagrangian deformations”, Compos. Math., 141:6 (2005), 1602–1614 | DOI | MR | Zbl

[20] Garay M. D., “Stable moment mappings and singular Lagrangian fibrations”, Quart. J. Math., 56 (2005), 357–366 | DOI | MR | Zbl

[21] Garay M. D., “Vanishing cycles in complex symplectic geometry”, Moscow Math. J., 8:1, 73–90 | MR | Zbl

[22] Garay M. D., “Finiteness and constructibility in local analytic geometry”, Enseign. Math. (to appear)

[23] Garay M. D., “Perturbative expansions in quantum mechanics”, Ann. Inst. Fourier (to appear)

[24] Garay M. D., van Straten D., “On the topology of Lagrangian Milnor fibres”, Intern. Math. Res. Not., 2003, no. 35, 1933–1943 | DOI | MR | Zbl

[25] Gérardy S., “Le système de Hénon-Heiles”, Ann. Fac. Sci. Toulouse. Math. Ser. 6, 6:4 (1997), 633–652 | MR | Zbl

[26] Gibson C. G., Wirthmüller K., du Plessis A. A., Looijenga E. J. N., Topological stability of smooth mappings, Lect. Notes Math., 552, Springer, Berlin, 1976 | MR | Zbl

[27] Greuel G.-M., “Der Gauss–Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten”, Math. Ann., 214 (1975), 235–266 | DOI | MR | Zbl

[28] Grothendieck A., “On the de Rham cohomology of algebraic varieties”, Publ. Math. IHES, 29 (1966), 95–103 | MR

[29] Heisenberg W., “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen”, Ztschr. Phys., 33 (1925), 879–893 | DOI | Zbl

[30] Helffer B., Sjöstrand J., “Semiclassical analysis for Harper's equation. III: Cantor structure of the spectrum”, Mém. Soc. math. France, 39 (1989), 1–124 | MR | Zbl

[31] Houzel C., “Espaces analytiques relatifs et théorème de finitude”, Math. Ann., 205 (1973), 13–54 | DOI | MR | Zbl

[32] Houzel C., Schapira P., “Images directes de modules différentiels”, C. r. Acad. sci. Paris. Sér. 1, 298:18 (1984), 461–464 | MR | Zbl

[33] Kiehl R., Verdier J.-L., “Ein einfacher Beweis des Kohärenzsatzes von Grauert”, Math. Ann., 195 (1971), 24–50 | DOI | MR | Zbl

[34] Lê D. T., “Singularités isoleés des hypersurfaces complexes”, Acta sci. Vietnam., 7 (1972), 24–33

[35] Lê D. T., “Some remarks on relative monodromy”, Real and complex singularities, Oslo, 1976, ed. P. Holm, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 397–404 | MR

[36] Lefschetz S., L'analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1924 | Zbl

[37] Lesfari A., “Le système différentiel de Hénon–Heiles et les variétés de Prym”, Pacif. J. Math., 212:1 (2003), 125–132 | DOI | MR | Zbl

[38] Malgrange B., “Sommation des séries divergentes”, Expos. math., 13:2–3 (1995), 163–222 | MR | Zbl

[39] Milnor J., Singular points of complex hypersurfaces, Ann. Math. Stud., 61, Princeton Univ. Press, Princeton, NJ, 1968, 130 pp. | MR | Zbl

[40] Miranda E., San Vũ Ngọc, “A singular Poincaré lemma”, Intern. Math. Res. Not., 2005, no. 1, 27–45 | DOI | MR

[41] Moser J., “On the volume elements on a manifold”, Trans. Amer. Math. Soc., 120:2 (1965), 286–294 | DOI | MR | Zbl

[42] Palamodov V. P., “O kratnosti golomorfnogo otobrazheniya”, Funkts. analiz i ego pril., 1:3 (1967), 54–65 | MR | Zbl

[43] Pham F., “La descente des cols par les onglets de Lefschetz, avec vues sur Gauss–Manin”, Astérisque, 130 (1985), 11–47 | MR | Zbl

[44] Pham F., “Resurgence, quantized canonical transformations, and multi-instanton expansions”, Algebraic analysis, v. 2, eds. M. Kashiwara, T. Kawai, Acad. Press, Boston, MA, 1988, 699–726 | MR

[45] Polesello P., Schapira P., “Stacks of quantization-deformation modules on complex symplectic manifolds”, Intern. Math. Res. Not., 2004, no. 49, 2637–2664 | DOI | MR | Zbl

[46] Rüssmann H., “Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung”, Math. Ann., 154 (1964), 285–300 | DOI | MR | Zbl

[47] Sato M., Kawai T., Kashiwara M., “Microfunctions and pseudo-differential equations”, Hyperfunctions and pseudo-differential equations, Lect. Notes Math., 287, Springer, Berlin, 1973, 263–529 | MR

[48] Schneiders J.-P., “A coherence criterion for Fréchet modules”, Astérisque, 224 (1994), 99–113 | MR | Zbl

[49] Schwartz L., “Homomorphismes et applications complètement continues”, C. r. Acad. sci. Paris, 236 (1953), 2472–2473 | MR | Zbl

[50] Sebastiani M., “Preuve d'une conjecture de Brieskorn”, Manuscr. math., 2 (1970), 301–308 | DOI | MR | Zbl

[51] Serre J.-P., “Faisceaux algébriques cohérents”, Ann. Math. Ser. 2, 61 (1955), 197–278 | DOI | MR | Zbl

[52] Sevenheck C., Lagrangian singularities, Ph. D. Thesis, J. Gutenberg Univ., Mainz, 2003 | MR

[53] Sevenheck C., van Straten D., “Deformation of singular lagrangian subvarieties”, Math. Ann., 327:1 (2003), 79–102 | DOI | MR | Zbl

[54] Siegel C. L., “Über die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung”, Math. Ann., 128 (1954), 144–170 | DOI | MR | Zbl

[55] Siu Y. T., “Every Stein subvariety admits a Stein neighborhood”, Invent. math., 38:1 (1976), 89–100 | DOI | MR | Zbl

[56] Stolovitch L., “Singular complete integrability”, Publ. Math. IHES, 91 (2000), 133–210 | MR | Zbl

[57] van Straten D., “On the Betti numbers of the Milnor fibre of a certain class of hypersurface singularities”, Singularities, representation of algebras, and vector bundles, Lect. Notes Math., 1273, eds. G.-M. Greuel, G. Trautmann, Springer, Berlin, 1987, 203–220 | MR

[58] van Straten D., On the quantization problem, Unpubl., 1991, 19 pp.

[59] Vey J., “Sur le lemme de Morse”, Invent. math., 40:1 (1977), 1–9 | DOI | MR | Zbl

[60] Vey J., “Sur certains sytèmes dynamiques séparables”, Amer. J. Math., 100 (1978), 591–614 | DOI | MR | Zbl

[61] Zinn-Justin J., Quantum field theory and critical phenomena, Oxford Sci. Publ., Oxford Univ. Press, Oxford, 1993, 996 pp. | MR

[62] Zung N. T., “Convergence versus integrability in Birkhoff normal form”, Ann. Math. Ser. 2, 161:1 (2005), 141–156 | DOI | MR | Zbl