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

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Matematicheskie Zametki, 2001, Volume 69, Issue 4, Pages 483–514
DOI: https://doi.org/10.4213/mzm519 (Mi mzm519)

This article is cited in 14 scientific papers (total in 14 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

Full-text PDF (421 kB) Citations (14)

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

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.).

Received: 22.06.2000

English version:
Mathematical Notes, 2001, Volume 69, Issue 4, Pages 437–466
DOI: https://doi.org/10.1023/A:1010252029050

Bibliographic databases:

UDC: 517.9

Language: Russian

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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Linking options:

https://www.mathnet.ru/eng/mzm519

https://doi.org/10.4213/mzm519

https://www.mathnet.ru/eng/mzm/v69/i4/p483

This publication is cited in the following 14 articles:

A. I. Klevin, “Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential”, Theoret. and Math. Phys., 199:3 (2019), 849–863
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
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
A. M. Chebotarev, T. V. Tlyachev, “Normal forms, inner products, and Maslov indices of general multimode squeezings”, Math. Notes, 95:5 (2014), 721–737
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
V. V. Belov, V. A. Maksimov, “Semiclassical quantization of Bohr orbits in the helium atom”, Theoret. and Math. Phys., 151:2 (2007), 659–680
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
Albeverio, S, “On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods”, Asymptotic Analysis, 43:3 (2005), 171
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
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
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
S. E. Roganova, “Moduli Spaces of Maslov Complex Germs”, Math. Notes, 71:5 (2002), 684–691
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
Belov, VV, “On global variables of the action-angle and harmonic oscillator types in neighborhoods of isotropic tori”, Doklady Mathematics, 64:3 (2001), 430

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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