Russian Journal of Mathematical Physics

, Volume 20, Issue 3, pp 283–294 | Cite as

Secondary resonances in Penning traps. Non-lie symmetry algebras and quantum states

  • M. V. KarasevEmail author
  • E. M. Novikova


The Penning trap Hamiltonian (hyperbolic oscillator in a homogeneous magnetic field) is considered in the basic three-frequency resonance regime. We describe its non-Lie algebra of symmetries. By perturbing the homogeneous magnetic field, we discover that, for special directions of the perturbation, a secondary hyperbolic resonance appears in the trap. For corresponding secondary resonance algebra, we describe its non-Lie permutation relations and irreducible representations realized by ordinary differential operators. Under an additional (Ioffe) inhomogeneous perturbation of the magnetic field, we derive an effective Hamiltonian over the secondary symmetry algebra. In an irreducible representation, this Hamiltonian is a model second-order differential operator. The spectral asymptotics is derived, and an integral formula for the asymptotic eigenstates of the entire perturbed trap Hamiltonian is obtained via coherent states of the secondary symmetry algebra.


Irreducible Representation Coherent State Symmetry Algebra Resonance Oscillator Secondary Resonance 
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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Laboratory for Mathematical Methods in Natural SciencesNational Research University Higher School of EconomicsMoscowRussia

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