Hamiltonian dynamics and spectral theory for spin-oscillators

Abstract

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.

32 pages

References in corpus (4)

• Semitoric integrable systems on symplectic 4-manifolds
• A semiclassical study of the Jaynes-Cummings model
• Almost toric symplectic four-manifolds
• Classical and quantum integrability

Cited by in corpus (13)

• Integrable systems, symmetries and quantization
• Taylor series and twisting-index invariants of coupled spin-oscillators
• Generating hyperbolic singularities in completely integrable systems
• Symplectic classification of coupled angular momenta
• Vu Ngoc's Conjecture on focus-focus singular fibers with multiple pinched points
• Faithful Semitoric Systems
• The height invariant of a four-parameter semitoric system with two focus-focus singularities
• Semitoric systems of non-simple type
• Hamiltonian monodromy and Morse theory
• Constructions of b-semitoric systems
• Classical Bethe Ansatz and Normal Forms in the Jaynes-Cummings Model
• Classifying Toric and Semitoric Fans by Lifting Equations from ${\mathrm{SL}}_2(\mathbb{Z})$
• The $p$-adic Jaynes-Cummings model in symplectic geometry