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А.А. Гурченков, Д.В. Башкина, Н.Т. Вилисова
Energy spectrum in a problem of a quantum rotator
© A.A. Gurchenkov, D.V. Bashkina, N.T. Velisova
Bauman Moscow State Technical University, Moscow 105005, Russia
The study tested the spectrum of eigenvalues of quantum systems, in the classical limit
admitting the existence of quadratic ones in the momenta of the first integrals. For example,
we consider the problem of two-dimensional potential well of finite depth and quantum
rotator. We made a comparison of bifurcations in classical and quantum problems. The
research showed that the presence of the additional integral imposes a partial ban on the
existence of separatrix contours. We examined an algebraic structure of classical integrals,
which determines the possibility of bringing Hamiltonian function to Liouville type and
separation of variables in Hamilton — Jacobi equation, which entails the separation of
variables in Schrödinger equation.
Keywords
: dynamical systems, bifurcation, Hamiltonian systems.
REFERENCES
[1] Eleonskiy V.M., Kulagin N.E. Domennye granitsy, solitony i fazovye perekhody
[Domain walls, solitons, and phase transitions].
Sb. Issledovaniya po fizike
kineticheskikh yavleniy
[Collected studies in physics of kinetic phenomena].
Sverdlovsk, Ufa Sci. Cen. Acad. Sci. USSR, 1984, pp. 56–65.
[2] Eleonskiy V.M., Kulagin N.E.
ZhETF — Journal of Experimental and Theoretical
Physics,
1983, vol. 84, iss. 2, pp. 616–628.
[3] Eleonskiy V.M., Kulagin N.E.
ZhETF — Journal of Experimental and Theoretical
Physics,
1987, vol. 93, iss. 4 (10), pp. 1436–1446.
[4] Prudnikov A.P., Brychkov Yu.A, Marychev O.I.
Integraly i ryady. Elementarnye
funktsii
[Integrals and series. Elementary functions]. Moscow, Nauka Publ., 1981,
800 p.
[5] Abromovits M., Stigan I., eds.
Spravochnik po spetsial’nym funktsiyam
[Handbook
of Mathematical Functions]. Moscow, Nauka Publ., 1979, 832 p.
[6] Beitmen G., Erdeii A.
Vysshie transtsendentnye funktsii. Ellipticheskie
i avtomorfnye funktsii. Funktsii Lame i Mat’e
[Higher transcendental functions.
Elliptic and automorphic functions. Lame and Mathieu functions]. Moscow, Nauka
Publ., 1967, 300 p.
[7] Yanke E., Emde F., Lesh F.
Spetsial’nye funktsii. Formuly, grafiki, tablitsy
[Special
functions. Formulas, graphs, tables]. Moscow, Nauka Publ., 1977, 344 p.
[8] Gurchenkov A.A., Esenkov A.S., Tsurkov V.I.
Izvestiya RAN. Teoriya i sistemy
upravleniya — Proceedings of the Russian Academy of Sciences. Control Theory
and Systems,
2006, no. 3, pp. 82–89.
[9] Gurchenkov A.A., Esenkov A.S., Tsurkov V.I.
Izvestiya RAN. Teoriya i sistemy
upravleniya — Proceedings of the Russian Academy of Sciences. Control Theory
and Systems,
2006, no. 1, pp. 141–148.
[10] Gurchenkov A.A., Esenkov A.S., Tsurkov V.I.
Avtomatika i telemekhanika —
Automatics and Telemechanics,
2007, no. 2, pp. 81–94.
[11] Gurchenkov A.A., Nosov M.V, Tsurkov V.I.
Control of Fluid-Containing Rotating
Rigid Bodies.
CRC Press, 2013, 147 p.
