Recent advances in the monodromy theory of integrable Hamiltonian systems

Martynchuk, Nikolay; Broer, Hendrik; Efstathiou, Konstantinos

doi:10.1016/j.indag.2020.05.001

Cited by 5 publications

(14 citation statements)

References 91 publications

(194 reference statements)

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“…Theorem 2 has been proved in Ref. 20 in the context of scattering problems. Here, we prove the result by explicitly constructing the symplectomorphism.…”

Section: Laplace‐runge‐lenz and C Neumannmentioning

confidence: 95%

“…From Ref. 20, we know that this system must be equivalent to the degenerate C. Neumann system on

T^{*} S^{2}

, and in Theorem 5 we give the explicit symplectomorphism that establishes this equivalence. Thus, Theorem 6 is already known indirectly, because using Theorem 5 the monodromy in the degenerate C. Neumann system applies.…”

Section: Introductionmentioning

confidence: 92%

“…The fact that the free particle when treated in this way has monodromy was first observed in Refs. 19, 20 as a special limit of Euler's two‐center problem in which the masses of both centers vanish. The main point of Refs.…”

Section: Introductionmentioning

confidence: 99%

“…The main point of Refs. 19, 20 was the investigation of scattering monodromy, and in Ref. 20 it was shown that after reduction by the free flow the degenerate C. Neumann system aka the quadratic spherical pendulum is obtained.…”

Section: Introductionmentioning

confidence: 99%

“…19, 20 was the investigation of scattering monodromy, and in Ref. 20 it was shown that after reduction by the free flow the degenerate C. Neumann system aka the quadratic spherical pendulum is obtained. The presence of monodromy in the quadratic spherical pendulum is well known, 21,22 and the interpretation as a degeneration of the C. Neumann system was elucidated in Ref.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Monodromy in prolate spheroidal harmonics

2021

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We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on L2false(S2false) have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

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“…Theorem 2 has been proved in Ref. 20 in the context of scattering problems. Here, we prove the result by explicitly constructing the symplectomorphism.…”

Section: Laplace‐runge‐lenz and C Neumannmentioning

confidence: 95%

“…From Ref. 20, we know that this system must be equivalent to the degenerate C. Neumann system on

T^{*} S^{2}

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Monodromy in prolate spheroidal harmonics

2021

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An introduction to classical monodromy: Applications to molecules in external fields

Omiste

González‐Férez

Ortega

2022

Journal of Mathematical Physics

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An integrable Hamiltonian system presents monodromy if the action-angle variables cannot be defined globally. As a prototype of classical monodromy with azimuthal symmetry, we consider a linear molecule interacting with external fields and explore the topology structure of its phase space. Based on the behavior of closed orbits around singular points or regions of the energy–momentum plane, a semi-theoretical method is derived to detect classical monodromy. The validity of the monodromy test is numerically illustrated for several systems with azimuthal symmetry.

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Hamiltonian Monodromy via spectral Lax pairs

Gutierrez Guillen,

Sugny,

Mardešić

2024

Journal of Mathematical Physics

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Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by a spectral Lax pair approach. From the Lax pair, we derive a Riemann surface which allows us to compute in a straightforward way the corresponding Monodromy matrix. The general results are applied to the Jaynes–Cummings model and the spherical pendulum.

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Recent advances in the monodromy theory of integrable Hamiltonian systems

Cited by 5 publications

References 91 publications

Monodromy in prolate spheroidal harmonics

Monodromy in prolate spheroidal harmonics

An introduction to classical monodromy: Applications to molecules in external fields

Hamiltonian Monodromy via spectral Lax pairs

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