Communications in Mathematical Physics

, Volume 309, Issue 1, pp 123–154 | Cite as

Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators



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.


Spectral Theory Toeplitz Operator Pseudodifferential Operator Hamiltonian Vector Hamiltonian Dynamics 
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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Mathematics DepartmentWashington UniversitySt. LouisUSA
  3. 3.Institut de Recherches Mathématiques de RennesUniversité de Rennes 1Rennes CedexFrance

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