A Lagrangian Fibration of the Isotropic 3-Dimensional Harmonic Oscillator with Monodromy

I. Chiscop, H. R. Dullin, K. Efstathiou, H. Waalkens

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Abstract

The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three Poisson commuting integrals and, correspondingly, three commuting operators, one of which is the Hamiltonian. We show that the Lagrangian fibration defined by the Hamiltonian, the z component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has a non-degenerate focus-focus point, and hence, non-trivial Hamiltonian monodromy for sufficiently large energies. The joint spectrum defined by the corresponding commuting quantum operators has non-trivial quantum monodromy implying that one cannot globally assign quantum numbers to the joint spectrum. Published under license by AIP Publishing.
Original languageEnglish
Article number032103
Number of pages15
JournalJournal of Mathematical Physics
Volume60
Issue number3
DOIs
Publication statusPublished - Mar-2019

Keywords

  • QUANTUM PHASE-TRANSITIONS
  • HAMILTONIAN-SYSTEMS
  • HYDROGEN-ATOM
  • PERTURBATIONS
  • CLASSIFICATION
  • NORMALIZATION
  • RESONANCE

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