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hal-00480299, version 1

Hamiltonian dynamics and spectral theory for spin-oscillators

Alvaro Pelayo 1, San Vu Ngoc () 2

Communications in Mathematical Physics 309, 1 (2012) 123-154

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.

  • 1:  Mathematics Department
  • University of California, Berkeley
  • 2:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • Domain : Mathematics/Symplectic Geometry
    Mathematics/Spectral Theory
    Mathematics/Dynamical Systems
  • Keywords : integrable systems – spin – oscillator – quantization – joint spectrum – invariants
  • Comment : 32 pages
 
  • hal-00480299, version 1
  • http://hal.archives-ouvertes.fr/hal-00480299
  • oai:hal.archives-ouvertes.fr:hal-00480299
  • From: San Vu Ngoc <>
  • Submitted on: Monday, 3 May 2010 22:38:59
  • Updated on: Friday, 1 June 2012 11:17:54