Singular Lagrangian manifolds and semiclassical analysis. (English) Zbl 1074.53066

Colin de Verdière, Yves

Duke Math. J. 116, No. 2, 263-298 (2003).

The theory of singularities of Lagrangian manifolds was initiated by V. I. Arnol’d [Funct. Anal. Appl. 15, 235–246 (1982; Zbl 0487.58003)], A. B. Givental’ [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 33, 55–112 (1988; Zbl 0731.58023)], and others. Lagrangian submanifolds of symplectic manifolds play important role in classical mechanics and microlocal analysis. The author and S. Vu Ngoc [Ann. Sci. Éc. Norm. Supér., IV. Sér. 36, No.1, 1-55 (2003; Zbl 1028.81026)] gave Bohr-Sommerfeld rules for semiclassical completely integrable systems with two degrees of freedom with non-degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non-singular. In this paper, the author extends these results to more singular systems and considers classical and semiclassical normal forms for singular Lagrangian manifolds with two-dimensional phase spaces in the real analytic category. The semiclassical normal forms give the analog of the WKB-ansatz for these systems, from which the singular Bohr-Sommerfeld rules can be derived. Furthermore, the author treats the case of the saddle-node bifurcation in greater detail.

Reviewer: Andrew Bucki (Edmond)

MSC:

53D12	Lagrangian submanifolds; Maslov index
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
35P20	Asymptotic distributions of eigenvalues in context of PDEs
58J37	Perturbations of PDEs on manifolds; asymptotics

Keywords:

singular Lagrangian manifold; infinitesimal deformation; normal form; WKB-ansatz

Citations:

Zbl 1028.81026; Zbl 0487.58003; Zbl 0731.58023

References:

[1]	V. I. Arnold, Lagrangian manifolds with singularities, asymptotic rays and the open swallowtail , Funct. Anal. Appl. 15 , no. 4 (1981), 235–246. · Zbl 0487.58003
[2]	——–, Mathematical Methods of Classical Mechanics , 2d ed., Grad. Texts in Math. 60 , Springer, New York, 1989.
[3]	–. –. –. –., “First steps of local symplectic algebra” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications: D. B. Fuchs’ 60th Anniversary Collection , Amer. Math. Soc. Transl. Ser. 2 194 , Amer. Math. Soc., Providence, 1999, 1–8. · Zbl 0921.00044
[4]	V. I. Arnold, S. M. Goussein-Zadé, and A. N. Varchenko, Singularités des applications différentiables, I , Mir, Moscow, 1986.
[5]	V. I. Arnold, V. A. Vasilev, V. V. Goryunov, and O. V. Lyashko, Dynamical Systems, VI , Encyclopedia Math. Sci. 6 , Springer, Berlin, 1993.
[6]	W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations , Universitext, Springer, New York, 2000. · Zbl 0942.34004
[7]	S. Bates and A. Weinstein, Lectures on the Geometry of Quantization , Berkeley Math. Lect. Notes 8 , Amer. Math. Soc., Providence, 1997. · Zbl 1049.53061
[8]	P. J. Braam and J. J. Duistermaat [H. Duistermaat], Normal forms of real symmetric systems with multiplicity , Indag. Math. (N.S.) 4 (1993), 407–421. · Zbl 0802.35176
[9]	E. Caliceti, S. Graffi, and M. Maioli, Perturbation of odd anharmonic oscillators , Comm. Math. Phys. 75 (1980), 51–66. · Zbl 0446.47044
[10]	E. Caliceti and M. Maioli, Odd anharmonic oscillators and shape resonances , Ann. Inst. H. Poincaré Sect. A (N. S.) 38 (1983), 175–186. · Zbl 0521.47009
[11]	Y. Colin de Verdière, M. Lombardi, and J. Pollet, The microlocal Landau-Zener formula , Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 95–127. · Zbl 0986.81027
[12]	Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique, I: Concentration microlocale , Comm. Partial Differential Equations 19 (1994), 1535–1563. · Zbl 0819.35116
[13]	–. –. –. –., Équilibre instable en régime semi-classique, II: Conditions de Bohr-Sommerfeld , Ann. Inst. H. Poincaré Phys. Théor. 61 (1994), 347–367.
[14]	–. –. –. –., Singular Bohr-Sommerfeld rules , Comm. Math. Phys. 205 (1999), 459–500. · Zbl 1157.81310
[15]	Y. Colin de Verdière and San Vũ Ng\?c, Singular Bohr-Sommerfeld rules for 2D integrable systems , to appear in Ann. Sci. École Norm. Sup. (4). · Zbl 1028.81026
[16]	Y. Colin de Verdière and J. Vey, Le lemme de Morse isochore , Topology 18 (1979), 283–293. · Zbl 0441.58003
[17]	E. Delabaere, H. Dillinger, and F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques , Ann. Institut Fourier (Grenoble) 43 (1993), 163–199. · Zbl 0766.34032
[18]	–. –. –. –., Exact semiclassical expansions for one-dimensional quantum oscillators , J. Math. Phys. 38 (1997), 6126–6184. · Zbl 0896.34051
[19]	E. Delabaere and F. Pham, Unfolding the quartic oscillator , Ann. Physics 261 (1997), 180–218. · Zbl 0977.34052
[20]	E. Delabaere and D. T. Trinh, Spectral analysis of the complex cubic oscillator , J. Phys. A 33 (2000), 8771–8796. · Zbl 1044.81555
[21]	R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation , Academic Press, London, 1973. · Zbl 0279.41030
[22]	J. J. Duistermaat [H. Duistermaat], Fourier Integral Operators , Progr. Math. 130 , Birkhäuser, Boston, 1996.
[23]	L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals: Elliptic case , Comment. Math. Helv. 65 (1990), 4–35. · Zbl 0702.58024
[24]	A. T. Fomenko, “Topological classification of all integrable Hamiltonian differential equations of general type with two degrees of freedom” in The Geometry of Hamiltonian Systems (Berkeley, 1989) , Math. Sci. Res. Inst. Publ. 22 , Springer, New York, 1991, 131–339. · Zbl 0753.58014
[25]	A. B. Givental’, Singular Lagrangian manifolds and their Lagrangian maps , J. Soviet Math. 52 (1990), 3246–3278. · Zbl 0900.58013
[26]	L. Hörmander, Fourier integral operators, I , Acta Math. 127 (1971), 79–183. · Zbl 0212.46601
[27]	——–, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators , Grundlehren Math. Wiss. 274 , Springer, Berlin, 1985. · Zbl 0601.35001
[28]	——–, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators , Grundlehren Math. Wiss. 275 , Springer, Berlin, 1985. · Zbl 0612.35001
[29]	B. Malgrange, Intégrales asymptotiques et monodromie , Ann. Sci. École Norm. Sup. (4) 7 (1974), 405–430. · Zbl 0305.32008
[30]	V. P. Maslov, Théorie des perturbations et méthodes asymptotiques , with appendices by V. I. Arnold and V. S. Busalaev, Dunod, Paris, 1972.
[31]	J. Milnor, Singular Points of Complex Hypersurfaces , Ann. of Math. Stud. 61 , Princeton Univ. Press, Princeton, 1968. · Zbl 0184.48405
[32]	Nguyen Hu’u Du’c and F. Pham, Germes de configurations legendriennes stables et fonctions d’Airy-Weber généralisées , Ann. Inst. Fourier (Grenoble) 41 (1991), 905–936. · Zbl 0741.58048
[33]	Nguyen Tien Zung [Zung, Nguyen Tien], A note on degenerate corank-one singularities of integrable Hamiltonian systems , Comment. Math. Helv. 75 (2000), 271–283. · Zbl 0961.37019
[34]	F. Pham, “Multiple turning points in exact WKB analysis (variations on a theme of Stokes)” in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear (Kyoto, 1998) , Kyoto Univ. Press, Kyoto, 2000, 71–85. · Zbl 1017.34091
[35]	J.-P. Ramis, Séries divergentes et théories asymptotiques , Bull. Soc. Math. France 121 (1993), Panor. Synth., supplement.
[36]	K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen , Invent. Math. 14 (1971), 123–142. · Zbl 0224.32011
[37]	San Vũ Ng\?c [Vũ Ng\?c, San], Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type , Comm. Pure Appl. Math. 53 (2000), 143–217. · Zbl 1027.81012
[38]	–. –. –. –., Formes normales semi-classiques des systèmes complètement intégrables au voisinage d’un point critique de l’application moment , Asymptot. Anal. 24 (2000), 319–342. · Zbl 0990.58018
[39]	R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem , J. Differential Geom. 58 (2001), 1–86. \CMP1 895 348 · Zbl 1052.53056
[40]	Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation , Transl. Math. Momogr. 82 , Amer. Math. Soc., Providence, 1990. · Zbl 1145.34378
[41]	D. van Straten and C. Sevenheck, Deformation of singular Lagrangian subvarieties , · Zbl 1051.14006
[42]	A. Voros, The return of the quartic oscillator: The complex WKB method , Ann. Inst. H. Poincaré Sect. A (N. S.) 39 (1983), 211–338. · Zbl 0526.34046
[43]	–. –. –. –., Exact resolution method for general 1D polynomial Schrödinger equation , J. Phys. A 32 (1999), 5993–6007. · Zbl 0967.81013