Hamiltonian dynamics and spectral theory for spin-oscillators

Details Hamiltonian dynamics and spectral theory for spin-oscillators Hamiltonian dynamics and spectral theory for spin-oscillators

2010-05-04

arxiv.org/abs/1005.0439v1

Mathematics

Symplectic Geometry

32 pages

Scientific paper

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.

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