A Lagrangian fibration of the isotropic 3-dimensional harmonic oscillator with monodromy

Chiscop, Irina; Dullin, Holger R.; Efstathiou, Konstantinos; Waalkens, Holger

doi:10.1063/1.5053887

Cited by 4 publications

(2 citation statements)

References 54 publications

(39 reference statements)

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“…In two recent papers, 17,18 we have used separation in spheroidal coordinates for the Kepler problem in space and the harmonic oscillator in space, respectively, and shown that both problems—when considered in prolate spheroidal variables—have Hamiltonian and quantum monodromy. The present paper grew out of the realization that an even simpler problem, namely, the free particle, can be studied in a similar way, and leads to similar results, namely, monodromy in the joint spectrum.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Monodromy in prolate spheroidal harmonics

2021

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“…In two recent papers [DW18] and [CDEW19] we have used separation in spheroidal coordinates for the Kepler problem in space and the harmonic oscillator in space, respectively, and shown that both problems -when considered in prolate spheroidal variables -have Hamiltonian and quantum monodromy. The present paper grew out of the realisation that an even simpler problem, namely the free particle, can be studied in a similar vain, and leads to similar results, namely monodromy in the joint spectrum.…”

Section: Introductionmentioning

confidence: 99%

Monodromy in Prolate Spheroidal Harmonics

Dawson,

Dullin,

Nguyen

2020

Preprint

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We show that spheroidal wave functions viewed as the essential part of the joint eigenfunction of two commuting operators of L2(S 2 ) has a defect in the joint spectrum that makes a global labelling 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 analogue 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 semi-classical limit of the quantum integrable system of commuting operators. We show that this is a semi-toric system with a non-degenerate focus-focus point, such that there is monodromy in the classical and the quantum system.

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Semitoric Families

Le Floch,

Palmer

2024

Memoirs of the AMS

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Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global S 1 \mathbb {S}^1 -action; these systems were classified by Pelayo and Vũ Ngọc in terms of five symplectic invariants. We introduce and study semitoric families, which are one-parameter families of integrable systems with a fixed S 1 \mathbb {S}^1 -action that are semitoric for all but finitely many values of the parameter, with the goal of developing a strategy to find a semitoric system associated to a given partial list of semitoric invariants. We also enumerate the possible behaviors of such families at the parameter values for which they are not semitoric, providing examples illustrating nearly all possible behaviors, which describes the possible limits of semitoric systems with a fixed S 1 \mathbb {S}^1 -action. Furthermore, we introduce natural notions of blowup and blowdown in this context, investigate how semitoric families behave under these operations, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants; these families are related to the semitoric minimal model program. Finally, we give several explicit semitoric families on the first and second Hirzebruch surfaces showcasing various possible behaviors of such families which include new semitoric systems.

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A Lagrangian fibration of the isotropic 3-dimensional harmonic oscillator with monodromy

Cited by 4 publications

References 54 publications

Monodromy in prolate spheroidal harmonics

Monodromy in prolate spheroidal harmonics

Monodromy in Prolate Spheroidal Harmonics

Semitoric Families

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