Qualitative Analysis of the Classical and Quantum Manakov Top
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Manakov top; energy-momentum diagram; monodromy.
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Evguenii Sinitsyn; Boris Zhilinskii. Qualitative Analysis of the Classical and Quantum Manakov Top. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a45/

[1] 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 | DOI | MR

[2] Audin M., Spinning tops, Chapter 4, Cambridge University Press, Cambridge, 1996 | MR | Zbl

[3] Bolsinov A. V., Fomenko A. T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, London, 2004, Section 14 | MR | Zbl

[4] 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 | DOI | MR | Zbl

[5] Child M. S., Quantum monodromy and molecular spectroscopy, Adv. Chem. Phys., to appear

[6] Colin de Verdière Y., Vũ Ngọc S., “Singular Bohr–Sommerfeld rules for 2D integrable systems”, Ann. Sci. Ècole Norm. Sup. (4), 36 (2003), 1–55 ; math.AP/0005264 | MR | Zbl

[7] Cushman R. H., Bates L. M., Global aspects of classical integrable systems, Birkhäuser, Basel, 1997 | MR | Zbl

[8] Cushman R. H., Sadovskii D., “Monodromy in the hydrogen atom in crossed fields”, Phys. D, 142 (2000), 166–196 | DOI | MR | Zbl

[9] Davison C. M., Dullin H. R., Bolsinov A. V., Geodesics on the ellipsoid and monodromy, math-ph/0609073 | MR

[10] Davison C. M., Dullin H. R., Geodesic flow on three dimensional ellipsoids with equal semi-axes, math-ph/0611060 | MR

[11] Duistermaat J. J., “On global action angle coordinates”, Comm. Pure Appl. Math., 33 (1980), 687–706 | DOI | MR | Zbl

[12] Efstathiou K., Cushman R. H., Sadovskii D. A., “Fractional monodromy in the $1:-2$ resonance”, Adv. Math., 209 (2007), 241–273 | DOI | MR | Zbl

[13] 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 | DOI | MR | Zbl

[14] 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 pp., ages | DOI

[15] 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 | DOI | MR | Zbl

[16] Komarov I. V., Kuznetsov V. B., “Quantum Euler–Manakov top on the 3-sphere $S_3$”, J. Phys. A: Math. Gen., 24 (1991), L737–L742 | DOI | MR | Zbl

[17] Leung N. C., Symington M.,, Almost toric symplectic four-manifolds, math.SG/0312165 | MR

[18] Manakov S. V., “Note on the integration of Euler's equation of the dynamics of an $N$ dimensional rigid body”, Funct. Anal. Appl., 10:4 (1976), 328–329 | DOI | MR

[19] Michel L., “Points critique des fonctions invariantes sur une $G$-varieté”, C. R. Math. Acad. Sci. Paris, 272 (1971), 433–436 | MR | Zbl

[20] Michel L., Zhilinskii B. I., “Symmetry, invariants, and topology. I. Basic tools”, Phys. Rep., 341 (2001), 11–84 | DOI | MR | Zbl

[21] Ne\t{kh}oroshev N. N., “Action-angle variables and their generalizations”, Tr. Mosk. Mat. Obs., 26, 1972, 180–198 | MR

[22] Ne\t{kh}oroshev 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 | MR

[23] Ne\t{kh}oroshev N. N., Sadovskii D., Zhilinskii B., “Fractional Hamiltonian monodromy”, Ann. Henri Poincaré, 7 (2006), 1099–1211 | DOI | MR

[24] Oshemkov A. A., “Topology of isoenergy surfaces and bifurcation diagrams for integrable cases of rigid body dynamics on $so(4)$”, Uspekhi Mat. Nauk, 42:6 (1987), 199–200 | MR

[25] Perelomov A. .M., Motion of four-dimensional rigid body around a fixed point: an elementary approach. I, math-ph/0502053 | MR

[26] Sadovskii D., Zhilinskii B.,, “Group theoretical and topological analysis of localized vibration-rotation states”, Phys. Rev. A, 47 (1993), 2653–2671 | DOI

[27] Sadovskii D., Zhilinskii B., “Monodromy, diabolic points, and angular momentum coupling”, Phys. Lett. A, 256 (1999), 235–244 | DOI | MR

[28] Sadovskii D., Zhilinskii B., “Quantum monodromy, its generalizations and molecular manifestations”, Mol. Phys., 104 (2006), 2595–2615 | DOI

[29] Sadovskii D., Zhilinskii B., “Hamiltonian systems with detuned $1:1:2$ resonance. Manifestations of bidromy”, Ann. Physics, 322 (2007), 164–200 | DOI | MR

[30] Symington M., “Four dimensions from two in symplectic topology”, Topology and Geometry of Manifolds (2001, Athens, GA), Proc. Symp. Pure Math., 71, AMS, Providence, RI, 2003, 153–208 ; math.SG/0210033 | MR | Zbl

[31] 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 | DOI

[32] Vũ Ngọc S., “Quantum monodromy in integrable systems”, Comm. Math. Phys., 203 (1999), 465–479 | DOI | MR | Zbl

[33] Vũ Ngọc S., “Moment polytopes for symplectic manifolds with monodromy”, Adv. Math., 208 (2007), 909–934 ; math.SG/0504165 | DOI | MR | Zbl

[34] Zhilinskii B. I., “Symmetry, invariants, and topology. II. Symmetry, invariants, and topology in molecular models”, Phys. Rep., 341 (2001), 85–171 | DOI | MR | Zbl

[35] Zhilinskii B., “Interpretation of quantum Hamiltonian monodromy in terms of lattice defects”, Acta Appl. Math., 87 (2005), 281–307 | DOI | MR | Zbl

[36] Zhilinskii B., “Hamiltonian monodromy as lattice defect”, Topology in Condensed Matter, ed. M. I. Monastyrsky, Springer, Berlin, 2006, 165–186 ; quant-ph/0303181 | MR | Zbl