Monodromy, diabolic points, and angular momentum coupling

Sadovskiı́, D.A.; Zhilinskii, B.I.

doi:10.1016/s0375-9601(99)00229-7

Cited by 126 publications

(134 citation statements)

References 13 publications

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“…Recent rigorous mathematical description of dynamical problems with such singularities can be found in [17,30,32,33]. More physically based intuitive construction was suggested in [14,27,35,36].…”

Section: Relation To Monodromy and Its Generalizationsmentioning

confidence: 99%

Qualitative Analysis of the Classical and Quantum Manakov Top

Sinitsyn¹,

Zhilinskii²

2007

SIGMA

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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. Personal introductionIt was about six years ago that I (B.Z.) met for the first time Vadim Kuznetsov during one of the 'Geometric Mechanics' conferences in Warwick University, U.K. I cannot say that we have found immediately mutual interest in our research works in spite of the fact that our problems were rather related. At that time I tried to understand better the manifestation of classical Hamiltonian monodromy in corresponding quantum problems and looked for different simple classical integrable models with monodromy which could be of interest for physical, mainly molecular, applications. Vadim worked on much more formal mathematical aspects of integrable models related to almost unknown for me special functions. I tried to convince him that from the point of view of physical applications the most important task is to understand qualitative features of integrable models using some simple geometric tools like classification of defects of regular lattices formed by joint spectrum of several commuting observables. Vadim insisted on special functions, complex analysis, Lie algebras etc. Nevertheless, we have found many points of common interest. Soon after, Vadim visited Dunkerque and we have tried to find some concrete problem, where we could demonstrate clearly what each of us means by understanding the solution. In fact, such problem was found quickly. It was the Manakov top. Vadim was interested in Manakov top because of its relation with XYZ Gaudin magnet. He published a short paper together with I. Komarov on this subject in 1991 [16]. For me the model problem like Manakov top represented certain interest because it is naturally related to molecular models constructed, for example, by coupled angular momenta or by angular momentum and

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Section: Relation To Monodromy and Its Generalizationsmentioning

confidence: 99%

Qualitative Analysis of the Classical and Quantum Manakov Top

Sinitsyn¹,

Zhilinskii²

2007

SIGMA

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“…Such foliations consist of a nontrivial bundle of two-dimensional regular Liouville tori and one singular surface, a pinched torus, see also [3], [19] or later in this paper. This type of foliation has been found in various two degree of freedom Hamiltonian systems, see for instance [3], [5], [13] and [14]. The most famous example is perhaps the spherical pendulum, see [6].…”

Section: Introductionmentioning

confidence: 86%

“…Monodromy in these KAM tori will again constitute an obvious obstruction to the global labeling of the quantum levels by two quantum numbers. The problems with the global labeling of quantum levels and the phenomenon of redistribution of quantum states have been studied extensively and have also been found experimentally, see for instance [5], [8] and [14].…”

Section: Introductionmentioning

confidence: 99%

A Cantor set of tori with monodromy near a focus–focus singularity

Rink

2003

Nonlinearity

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We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well-known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantisation rules select sequences of KAM tori that correspond to quantum levels. Hence a global labeling of quantum levels by two quantum numbers is not possible.

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“…Typically, singular values of a completely integrable system are responsible for the topology of the torus bundle [7], and the topology of the torus bundle has remarkable consequences on the distribution of quantum energy levels of the associated quantum system [22,12]. It follows that the global analysis of, and the quantum mechanics associated to a completely integrable system rely on its singularities.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

Giacobbe

2007

Regul. Chaot. Dyn.

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In this article we give a list of 10 rank zero and 6 rank one singularities of 2 degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system. Mathematics Subject Classification: 55R55 Fiberings with singularities, 37J35 Completely integrable systems, topological structure of phase space, integration methods.

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Monodromy, diabolic points, and angular momentum coupling

Cited by 126 publications

References 13 publications

Qualitative Analysis of the Classical and Quantum Manakov Top

Qualitative Analysis of the Classical and Quantum Manakov Top

A Cantor set of tori with monodromy near a focus–focus singularity

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

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