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Mat. Zametki, 2001, Volume 69, Issue 4, Pages 483–514 (Mi mz519)  

This article is cited in 12 scientific papers (total in 12 papers)

Isotropic Tori, Complex Germ and Maslov Index, Normal Forms and Quasimodes of Multidimensional Spectral Problems

V. V. Belov, O. S. Dobrokhotov, S. Yu. Dobrokhotov

Moscow State Institute of Electronics and Mathematics

Abstract: More than twenty years ago V. P. Maslov posed the question under what conditions it is possible to assign to invariant isotropic lower-dimensional tori of Hamiltonian systems sequences of asymptotic eigenvalues and eigenfunctions (spectral series) of the corresponding quantum mechanical and wave operators. In the present paper this question is answered in terms of the quadratic approximation to the theory of normal forms. We also discuss the quantization conditions for isotropic tori and their relation to topological, geometric, and dynamical characteristics (Maslov indices, rotation (winding) numbers, eigenvalues of dynamical flows, etc.).

DOI: https://doi.org/10.4213/mzm519

Full text: PDF file (421 kB)
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English version:
Mathematical Notes, 2001, 69:4, 437–466

Bibliographic databases:

UDC: 517.9
Received: 22.06.2000

Citation: V. V. Belov, O. S. Dobrokhotov, S. Yu. Dobrokhotov, “Isotropic Tori, Complex Germ and Maslov Index, Normal Forms and Quasimodes of Multidimensional Spectral Problems”, Mat. Zametki, 69:4 (2001), 483–514; Math. Notes, 69:4 (2001), 437–466

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Belov, V. A. Maksimov, “Semiclassical Spectral Series of a Helium-like Atom in a Magnetic Field”, Theoret. and Math. Phys., 126:3 (2001), 378–395  mathnet  crossref  crossref  zmath  isi
    2. Belov, VV, “On global variables of the action-angle and harmonic oscillator types in neighborhoods of isotropic tori”, Doklady Mathematics, 64:3 (2001), 430  zmath  adsnasa  isi
    3. J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashin, “The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field”, Theoret. and Math. Phys., 131:2 (2002), 704–728  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. S. E. Roganova, “Moduli Spaces of Maslov Complex Germs”, Math. Notes, 71:5 (2002), 684–691  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. V. Belov, S. Yu. Dobrokhotov, V. A. Maksimov, “Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application”, Theoret. and Math. Phys., 135:3 (2003), 765–791  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. S. Yu. Dobrokhotov, M. A. Poteryakhin, “Normal Forms near Two-Dimensional Resonance Tori for the Multidimensional Anharmonic Oscillator”, Math. Notes, 76:5 (2004), 653–664  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. M. A. Poteryakhin, “Normal forms near an invariant torus and the asymptotic eigenvalues of the operator $\langle V,\nabla\rangle-\epsilon\Delta$”, Math. Notes, 77:1 (2005), 140–145  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Albeverio, S, “On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods”, Asymptotic Analysis, 43:3 (2005), 171  mathscinet  zmath  isi  elib
    9. V. V. Belov, V. A. Maksimov, “Semiclassical quantization of Bohr orbits in the helium atom”, Theoret. and Math. Phys., 151:2 (2007), 659–680  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Kulikovskii A.G., Pashchenko N.T., “The Effect of a Small Background Inhomogeneity on the Asymptotic Properties of Linear Perturbations”, Pmm-J. Appl. Math. Mech., 74:2 (2010), 127–134  crossref  mathscinet  isi  scopus  scopus
    11. A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirozzi, “Scalarization of stationary semiclassical problems for systems of equations and its application in plasma physics”, Theoret. and Math. Phys., 193:3 (2017), 1761–1782  mathnet  crossref  crossref  adsnasa  isi  elib
    12. A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirozzi, “Gausian packets and beams with focal points in vector problems of plasma physics”, Theoret. and Math. Phys., 196:1 (2018), 1059–1081  mathnet  crossref  crossref  adsnasa  isi  elib
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