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Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton-ehrenfest system. (English. Russian original) Zbl 1118.81033
Theor. Math. Phys. 150, No. 1, 21-33 (2007); translation from Teor. Mat. Fiz. 150, No. 1, 26-40 (2007).
Summary: We consider the classical equations of motion in quantum means, i.e., the Hamilton-Ehrenfest system. In the semiclassical approximation in the framework of the covariant approach based on these equations, we construct the spectral series of a nonlinear Hartree-type operator corresponding to a rest point.

MSC:
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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[1] M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991); English transl., Amer. Math. Soc., Providence, R. I. (1993). · Zbl 0731.58002
[2] S. P. de Groot and L. G. Suttirp, Foundations of Electrodynamics, North-Holland, Amsterdam (1972).
[3] Y. Lai and H. A. Haus, Phys. Rev. A, 40, 844–853, 854–866 (1989).
[4] L. P. Pitaevskii, Phys. Usp., 41, 569–580 (1998).
[5] A. S. Davydov, Solitons in Molecular Systems [in Russian], Naukova Dumka, Kiev (1984); English transl., Reidel, Dordrecht (1985). · Zbl 0532.60028
[6] M. Born, Z. Phys., 38, 803–827 (1926); P. Ehrenfest, Z. Phys., 45, 455–457 (1927).
[7] V. P. Maslov, Theorie des perturbations et methodes asymptotiques [in Russian], Moscow State Univ., Moscow (1965); French transl., Dunod, Paris (1972); V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics [in Russian], Nauka, Moscow (1976); English transl., Kluwer, Dordrecht (1981).
[8] V. P. Maslov, Asymptotic Methods and Perturbation Theory [in Russian], Nauka, Moscow (1988). · Zbl 0653.35002
[9] V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods, Mir, Moscow (1976).
[10] V. P. Maslov, Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977); English transl.: ComplexWKB Method for Nonlinear Equations: I. Linear Theory, Birkhäuser, Basel (1994); V. V. Belov and S. Yu. Dobrokhotov, Theor. Math. Phys., 92, 843–868 (1992).
[11] M. V. Karasev and A. V. Pereskokov, Theor. Math. Phys., 79, 479–486 (1989); 97, 1160–1170 (1993); Izv. Math., 65, 883–921 (2001); 65, 1127–1168 (2001); M. V. Karasev and V. P. Maslov, J. Sov. Math., 15, 273–368 (1981).
[12] V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Int. J. Math. Math. Sci., 32, No. 6, 325–370 (2002). · Zbl 1136.81372
[13] V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Theor. Math. Phys., 130, 391–418 (2002). · Zbl 1044.81045
[14] A. L. Lisok, A. Yu. Trifonov, and A. V. Shapovalov, J. Phys. A, 37, 4535–4556 (2004); Theor. Math. Phys., 141, 1528–1541 (2004). · Zbl 1051.35060
[15] A. L. Lisok, A. Yu. Trifonov, and A. V. Shapovalov, Proc. Inst. Math. NAS Ukr., 50, 1454–1465 (2004); Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 1, 007 (2005); F. N. Litvinets, A. V. Shapovalov, and A. Yu. Trifonov, J. Phys. A, 39, 1191–1206 (2006).
[16] V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, Ann. Phys. (NY), 246, 231–280 (1996); ”Semiclassical concentrated states of the Schrödinger equation,” in: Lecture Notes in Theoretical and Mathematical Physics (A. V. Aminova, ed.), Vol. 1, Part 1, Izd-vo BOG, Kazan (1996), pp. 15–136. · Zbl 0874.35099
[17] V. V. Belov and M. F. Kondrat’eva, Math. Notes, 56, 1228–1236 (1994); 58, 1251–1261 (1995). · Zbl 0840.34089
[18] V. G. Bagrov, V. V. Belov, and M. F. Kondrat’eva, Theor. Math. Phys., 98, 34–38 (1994); V. G. Bagrov et al., J. Moscow Phys. Soc., 3, 309–320 (1993); ”The quasiclassical localization of the states and a new approach of quasi-classical approximation in quantum mechanics,” in: Particle Physics, Gauge Fields, and Astrophysics (A. I. Studenikin, ed.), Accademia Nazilonale dei Lincei, Rome (1994), pp. 132–142; V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, ”New methods for semiclassical approximation in quantum mechanics, ” in: Proc. Intl. Workshop ”Quantum Systems: New Trends and Methods” (A. O. Barut, I. D. Feranchuk, Ya. M. Shnir, and L. M. Tomil’chik, eds.), World Scientific, Singapore (1995), pp. 533–543.
[19] M. A. Malkin and V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979); A. M. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin (1986).
[20] V. G. Bagrov, V. V. Belov, and I. M. Ternov, Theor. Math. Phys., 50, 256–261 (1982); J. Math. Phys., 24, 2855–2859 (1983); V. V. Belov and V. P. Maslov, Sov. Phys. Dokl., 34, 220–223 (1989); 35, 330–332 (1990); V. G. Bagrov, V. V. Belov, and A. Yu. Trifonov, ”Semiclassical trajectory-coherent approximation in quantum mechanics: II. High order corrections to the Dirac operators in external electromagnetic field,” quant-ph/9806017 (1998); V. V. Belov and M. F. Kondrat’eva, Theor. Math. Phys., 92, 722–735 (1992); V. G. Bagrov, V. V. Belov, A. Yu. Trifonov, and A. A. Yevseyevich, Class. Q. Grav., 8, 515–527, 1349–1359, 1833–1846 (1991); V. G. Bagrov, A. Yu. Trifonov, and A. A. Yevseyevich, Class. Q. Grav., 9, 533–543 (1992).
[21] H. Bateman and A. Erd’elyi, eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 2, McGraw-Hill, New York (1953).
[22] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Plenum, New York (1984); M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981). · Zbl 0598.35002
[23] I. V. Simenog, Theor. Math. Phys., 30, 263–268 (1977).
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