Integrable systems, symmetries, and quantization

DOI10.1007/s11005-017-1018-zzbMath1390.37101arXiv1704.06686OpenAlexW2951302201MaRDI QIDQ1745921

Daniele Sepe, San Vũ Ngoc

Publication date: 18 April 2018

Published in: Letters in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1704.06686

zbMATH Keywords

quantum systemclassical integrable systemalmost-toric singular fibersemi-toric systemspectrum of quantum operators

Mathematics Subject Classification ID

Symplectic manifolds (general theory) (53D05) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Geometry and quantization, symplectic methods (81S10) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20) Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics (70H06) Commutation relations and statistics as related to quantum mechanics (general) (81S05) Geometric quantization (53D50)

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Cites Work

• Inverse spectral theory for semiclassical Jaynes-Cummings systems
• Normalisation of the smooth focus-focus: a simple proof
• Smooth normal forms for integrable Hamiltonian systems near a focus-focus singularity
• Semi-global symplectic invariants of the Euler top
• Twisted isotropic realisations of twisted Poisson structures
• First steps in symplectic and spectral theory of integrable systems
• From compact semi-toric systems to Hamiltonian \(S^1\)-spaces
• Constructing integrable systems of semitoric type
• The local structure of Poisson manifolds
• Hamiltonian dynamics and spectral theory for SPIN-oscillators
• On the Kostant multiplicity formula
• Semitoric integrable systems on symplectic 4-manifolds
• Moment polytopes for symplectic manifolds with monodromy
• Non-compact symplectic toric manifolds
• Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case
• Partial differential equations. II: Qualitative studies of linear equations
• Le problème général des variables actions-angles. (The general problem of action-angle variables)
• Le lemme de Morse isochore
• An obstruction to the existence of affine structures
• On the variation in the cohomology of the symplectic form of the reduced phase space
• Convexity properties of the moment mapping
• On action-angle variables
• A note on focus-focus singularities
• Non-Abelian convexity by symplectic cuts
• Berezin--Toeplitz operators, a semi-classical approach
• Symplectic linearization of singular Lagrangian foliations in \(M^{4}\).
• Another note on focus-focus singularities
• Global action-angle variables for non-commutative integrable systems
• Faithful semitoric systems
• Monodromy in two degrees of freedom integrable systems
• On semi-global invariants for focus-focus singularities
• An extension of the Reeb stability theorem
• Noncommutative integrability and action-angle variables in contact geometry
• Semi-global symplectic invariants of the spherical pendulum
• Monodromy, diabolic points, and angular momentum coupling
• Smooth invariants of focus-focus singularities and obstructions to product decomposition
• Holomorphic Morse inequalities and Bergman kernels
• Symplectic manifolds and their Lagrangian submanifolds
• Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung
• Contact isotropic realisations of Jacobi manifolds via Spencer operators
• Holonomy of flat affinely connected manifolds
• Symplectic inverse spectral theory for pseudodifferential operators
• Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators
• Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems
• Symplectic theory of completely integrable Hamiltonian systems
• Proper groupoids and momentum maps: Linearization, affinity, and convexity
• The quantum mechanical spherical pendulum
• Hamiltoniens périodiques et images convexes de l'application moment
• On global action-angle coordinates
• Symplectic geometry
• Convexity and Commuting Hamiltonians
• Sur Certains Systemes Dynamiques Separables
• Periodic Hamiltonian flows on four-dimensional manifolds
• Integrable Hamiltonian system with two degrees of freedom. The topological structure of saturated neighbourhoods of points of focus-focus and saddle-saddle type
• The symmetries of solitons
• The affine invariant of proper semitoric integrable systems
• Singular Bohr–Sommerfeld rules for 2D integrable systems
• Quantum states in a champagne bottle
• Focus-focus singularities in classical mechanics
• Isospectrality for quantum toric integrable systems
• Bounds for the Dimensions of Certain Lie Algebras
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