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.
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Communicated by S. Zelditch
Partly supported by NSF Postdoctoral Fellowship, NSF Grant DMS-0965738, and an Oberwolfach Leibniz Fellowship.
Partially supported by an ANR ’Programme Blanc’.
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Pelayo, Á., Vũ Ngọc, S. Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators. Commun. Math. Phys. 309, 123–154 (2012). https://doi.org/10.1007/s00220-011-1360-4
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DOI: https://doi.org/10.1007/s00220-011-1360-4