Abstract
Maslov complex germs (complex vector bundles, satisfying certain additional conditions, over isotropic submanifolds of the phase space) are one of the central objects in the theory of semiclassical quantization. To these bundles one assigns spectral series (quasimodes) of partial differential operators. We describe the moduli spaces of Maslov complex germs over a point and a closed trajectory and find the moduli of complex germs generated by a given symplectic connection over an invariant torus.
Similar content being viewed by others
REFERENCES
V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow, 1977.
V. P. Maslov and O. Yu. Shvedov, The Complex Germ Method in the Many-Particle Problem and in Quantum Field Theory [in Russian], URSS, Moscow, 2000.
V. M. Babich, “Eigenfunctions which are concentrated in the neighborhood of a closed geodesic,” Zap. Nauch. Sem. LOMI, 9 (1968), 15–63.
V. V. Belov and S. Yu. Dobrokhotov, “The Maslov canonical operator on isotropic manifolds with a complex germ and its applications to spectral problems,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 299 (1988), no. 5, 1037–1042.
V. V. Belov and S. Yu. Dobrokhotov, “Semiclassical Maslov asymptotics with complex phases. I. General approach,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 92 (1992), no. 2, 215–254.
V. V. Belov, O. S. Dobrokhotov, and S. Yu. Dobrokhotov, “Isotropic tori, Maslov's complex germ and index, normal forms, and quasimodes in multidimensional spectral problems,” Mat. Zametki [Math. Notes], 69 (2001), no. 4, 483–514.
Yu. M. Vorob?ev, “The complex Maslov germ generated by a linear connection,” Mat. Zametki [Math. Notes], 48 (1990), no. 6, 29–37.
Yu. M. Vorob?ev, “Quantization of isotropic tori in some nonintegrable cases,” Uspekhi Mat. Nauk [Russian Math. Surveys], 45 (1990), no. 4, 127.
Yu. M. Vorob?ev and V. A. Itskov, “Quasimodes that correspond to conditionally periodic motion of stable type,” Mat. Zametki [Math. Notes], 55 (1994), no. 5, 36–42.
S. Yu. Dobrokhotov and V. Martinez-Olive, “Closed trajectories and two-dimensional tori in a quantum spectral problem for a three-dimensional anharmonic oscillator,” Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.], 58 (1997), 4–87.
S. Yu. Dobrokhotov, V. Martinez-Olive, and A. I. Shafarevich, “Closed trajectories and two-dimensional tori in the quantum Kepler problem with variable anisotropy,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 355 (1997), no. 3, 299–302.
A. D. Krakhnov, “Asymptotic behavior of the eigenvalues of pseudodifferential operators and invariant tori,” Uspekhi Mat. Nauk [Russian Math. Surveys], 31 (1976), no. 3, 66–74.
Topological Methods in the Theory of Hamiltonian Systems (A. B. Bolsinov, A. T. Fomenko, A. I. Shafarevich, editors) [in Russian], Faktorial, Moscow, 1998.
S. Yu. Dobrokhotov, V. Martinez-Olive, and A. I. Shafarevich, “Closed trajectories and two-dimensional tori in the quantum problem for a three-dimensional resonant anharmonic oscillator,” Russ. J. Math. Phys., 3 (1995), no. 1, 133–138.
S. Yu. Dobrokhotov and A. I. Shafarevich, “Quantum selection in semi-classical approximation of isotropic tori in partially integrable Hamiltonian systems,” Russ. J. Math. Phys., 5 (1998), no. 2, 267–272.
J. T. Duistermaat and V. W. Guillemin, “The spectrum of positive operators and periodic bicharacteristics,” Inventiones Math., 29 (1975), 39–79.
V. W. Guillemin and A. Weinstein, “Eigenvalues associated with geodesics,” Bull. Amer. Math. Soc., 82 (1976), no. 1, 92–94.
M. V. Karasev and Yu. M. Vorobjev, Integral Representation over Isotropic Submanifolds and Equations of Zero Curvature, Preprint MIEM, 1992.
J. V. Ralston, “On the construction of quasimodes associated with stable periodic orbits,” Comm. Math. Phys., 51 (1976), 219–242.
A. Voros, “The WKB-Maslov method for non-separable systems,” in Géométrie symplectique et physique mathématique (Colloq. Internat. CNRS, no. 237, Aix-en-Provence, 1974), Éditions du CNRS, Paris, 1975, pp. 277–287.
S. Yu. Dobrokhotov, B. Valino-Alonso, and N. N. Nekhoroshev, “A complex germ in systems with one cyclic variable,” Uspekhi Mat. Nauk [Russian Math. Surveys], 39 (1984), no. 3, 233–234.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Roganova, S.E. Moduli Spaces of Maslov Complex Germs. Mathematical Notes 71, 684–691 (2002). https://doi.org/10.1023/A:1015844106564
Issue Date:
DOI: https://doi.org/10.1023/A:1015844106564