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SIGMA 3 (2007), 046, 23 pages math-ph/0703045
https://doi.org/10.3842/SIGMA.2007.046
Contribution to the Vadim Kuznetsov Memorial Issue
Qualitative Analysis of the Classical and Quantum Manakov Top
Evguenii Sinitsyn a and Boris Zhilinskii b
a) Physics Department, Tomsk State University, 634050 Tomsk, Russia
b) Université du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France
Received 20 October, 2006, in final form 19 January, 2007; Published online March 13, 2007
Abstract
Qualitative features of the Manakov top are discussed for the
classical and quantum versions of the problem. Energy-momentum
diagram for this integrable classical problem and quantum joint
spectrum of two commuting observables for associated quantum problem
are analyzed. It is demonstrated that the evolution of
the specially chosen quantum cell through the joint quantum spectrum
can be defined for paths which cross singular strata. The corresponding
quantum monodromy transformation is introduced.
Key words:
Manakov top; energy-momentum diagram; monodromy.
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References
- Adler M., van Moerbeke P.,
The Kowalewski and Hénon-Heiles motions as Manakov geodesic
flows on SO(4) - a two-dimensional family of Lax pairs,
Comm. Math. Phys. 113 (1988), 659-700.
- Audin M., Spinning tops, Cambridge University Press,
Cambridge, 1996, Chapter 4.
- Bolsinov A.V., Fomenko A.T., Integrable
Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC
London, 2004, Section 14.
- Cejnar P., Macek M., Heinze S., Jolie J.,
Dobes J., Monodromy and excited-state quantum phase transitions in
integrable systems: collective vibrations of nuclei, J. Phys. A: Math. Gen.
39 (2006), L515-L521.
- Child M.S., Quantum monodromy and molecular
spectroscopy, Adv. Chem. Phys., to appear.
- Colin de Verdière Y., Vu Ngoc S.,
Singular Bohr-Sommerfeld rules for 2D integrable systems,
Ann. Sci. Ècole Norm. Sup. (4) 36 (2003), 1-55, math.AP/0005264.
- Cushman R.H., Bates L.M.,
Global aspects of classical integrable systems,
Birkhäuser, Basel, 1997.
- Cushman R.H., Sadovskii D.,
Monodromy in the hydrogen atom in crossed fields,
Phys. D 142 (2000), 166-196.
- Davison C.M., Dullin H.R., Bolsinov A.V.,
Geodesics on the ellipsoid and monodromy,
math-ph/0609073.
- Davison C.M., Dullin H.R.,
Geodesic flow on three dimensional ellipsoids with equal
semi-axes, math-ph/0611060.
- Duistermaat J.J.,
On global action angle coordinates,
Comm. Pure Appl. Math. 33 (1980), 687-706.
- Efstathiou K., Cushman R.H., Sadovskii D.A.,
Fractional monodromy in the 1:-2 resonance, Adv. Math.
209 (2007), 241-273.
- Efstathiou K., Sadovskii D., Zhilinskii B.,
Analysis of rotation-vibration relative equilibria on the
example of a tetrahedral four atom molecule,
SIAM J. Appl. Dyn. Syst. 3 (2004), 261-351.
- Grondin L., Sadovskii D., Zhilinskii B.,
Monodromy in systems with coupled angular momenta and rearrangement
of bands in quantum spectra, Phys. Rev. A 65 (2002), 012105, 15 pages.
- Kalnins E. G., Miller W.Jr., Winternitz P.,
The group O(4), separation of variables and the hydrogen atom,
SIAM J. Appl. Math. 30 (1976), 630-664.
- Komarov I.V., Kuznetsov V.B.,
Quantum Euler-Manakov top on the 3-sphere S3,
J. Phys. A: Math. Gen. 24 (1991), L737-L742.
- Leung N.C., Symington M.,
Almost toric symplectic four-manifolds, math.SG/0312165.
- Manakov S.V.,
Note on the integration of Euler's equation of the dynamics of an
N dimensional rigid body,
Funct. Anal. Appl. 11 (1976), 328-329.
- Michel L., Points critique des fonctions invariantes
sur une G-varieté, C. R. Math. Acad. Sci. Paris 272 (1971), 433-436.
- Michel L., Zhilinskii B.I.,
Symmetry, invariants, and topology. I. Basic tools,
Phys. Rep. 341 (2001), 11-84.
- Nekhoroshev N.N.,
Action-angle variables and their generalizations,
Tr. Mosk. Mat. Obs. 26 (1972), 180-198.
- Nekhoroshev N. N., Sadovskií D.A., Zhilinskií B.I.,
Fractional monodromy of resonant classical and quantum oscillators,
C. R. Math. Acad. Sci. Paris 335 (2002), 985-988.
- Nekhoroshev N. N., Sadovskii D., Zhilinskii B.,
Fractional Hamiltonian monodromy, Ann. Henri Poincaré
7 (2006), 1099-1211.
- Oshemkov A.A., Topology of isoenergy surfaces and
bifurcation diagrams for integrable cases of rigid body dynamics on
so(4), Uspekhi Mat. Nauk 42 (1987), 199-200.
- Perelomov A.M., Motion of four-dimensional rigid body
around a fixed point: an elementary approach. I,
math-ph/0502053.
- Sadovskii D., Zhilinskii B.,
Group theoretical and topological analysis of localized
vibration-rotation states,
Phys. Rev. A 47 (1993), 2653-2671.
- Sadovskii D., Zhilinskii B.,
Monodromy, diabolic points, and angular momentum
coupling, Phys. Lett. A 256 (1999), 235-244.
- Sadovskii D., Zhilinskii B.,
Quantum monodromy, its generalizations and molecular manifestations,
Mol. Phys. 104 (2006), 2595-2615.
- Sadovskii D., Zhilinskii B.,
Hamiltonian systems with detuned 1:1:2 resonance. Manifestations
of bidromy, Ann. Physics 322 (2007), 164-200.
- Symington M., Four dimensions from two in symplectic
topology, in Topology and Geometry of Manifolds (2001, Athens, GA), Proc. Symp. Pure Math., Vol. 71, AMS,
Providence, RI, 2003, 153-208, math.SG/0210033.
- Winnewisser M., Winnewisser B., Medvedev I.,
de Lucia F.C., Ross S.C., Bates L.M.,
The hidden kernel of molecular quasi-linearity: quantum monodromy,
J. Mol. Structure 798 (2006), 1-26.
- Vu Ngoc S.,
Quantum monodromy in integrable systems,
Comm. Math. Phys. 203 (1999), 465-479.
- Vu Ngoc S., Moment polytopes for symplectic
manifolds with monodromy, Adv. Math. 208 (2007), 909-934, math.SG/0504165.
- Zhilinskii B.I., Symmetry, invariants, and topology.
II Symmetry, invariants, and topology in molecular models,
Phys. Rep. 341 (2001), 85-171.
- Zhilinskii B., Interpretation of quantum
Hamiltonian monodromy in terms of lattice defects,
Acta Appl. Math. 87 (2005), 281-307.
- Zhilinskii B., Hamiltonian monodromy as
lattice defect, in Topology in Condensed Matter,
Editor M.I. Monastyrsky, Springer, Berlin, 2006, 165-186, quant-ph/0303181.
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