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Hamiltonian Dynamics and Spectral Theory for Spin–Oscillators

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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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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA

    Álvaro Pelayo

  2. Mathematics Department, Washington University, One Brookings Drive, Campus Box 1146, St. Louis, MO, 63130-489, USA

    Álvaro Pelayo

  3. Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    San Vũ Ngọc

Authors
  1. Álvaro Pelayo
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  2. San Vũ Ngọc
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Correspondence to Álvaro Pelayo.

Additional information

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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