ar X iv : h ep - t h / 04 06 18 4 v 1 2 2 Ju n 20 04 Quantum Oscillator on I CP n in a constant magnetic field

We construct the quantum oscillator interacting with a constant magnetic field on complex projective spaces I CP, as well as on their non-compact counterparts, i. e. the N−dimensional Lobachewski spaces LN . We find the spectrum of this system and the complete basis of wavefunctions. Surprisingly, the inclusion of a magnetic field does not yield any qualitative change in the energy spectrum. For N > 1 the magnetic field does not break the superintegrability of the system, whereas for N = 1 it preserves the exact solvability of the system. We extend this results to the cones constructed over I CP and LN , and perform the (Kustaanheimo-Stiefel) transformation of these systems to the three-dimensional Coulomb-like systems. Introduction The harmonic oscillator plays a fundamental role in quantum mechanics. On the other hand, there are few articles related with the oscillator on curved spaces. The most known generalization of the Euclidian oscillator is the oscillator on curved spaces with constant curvature (sphere and hyperboloid) [1] given by the potential VHiggs = ωr 0 2 x x20 , ǫx + x20 = r 2 0 , ǫ = ±1. (1) This system received much attention since its introduction (see for a review [2] and refs. therein) and is presently known under the name of “Higgs oscillator”. Recently the generalization of the oscillator to Kähler spaces has also been suggested, in terms of the potential [3] Vosc = ω g∂āK∂bK. (2) Various properties of the systems with this potential were studied in Refs. [3, 4, 5, 6]. It was shown that on the complex projective spaces I CP such a system inherits the whole set of rotational symmetries and a part of the hidden symmetries of the 2N−dimensional flat oscillator [3]. In Ref. [4], the classical solutions of the system on I CP, L2 (the noncompact counterpart of I CP) and the related cones were presented, and the reduction to three dimensions was studied. Particulary, it was found that the oscillator on some cone related with I CP (L2) results, after Hamiltonian reduction, in the Higgs oscillator on the three-dimensional sphere (two-sheet hyperboloid) in the presence of a Dirac monopole field. In Ref. [5] we presented the exact quantum mechanical solutions for the oscillator on I CP, L2 and related cones. We also reduced these quantum systems to three dimensions and performed their (Kustaanheimo-Stieffel) transformation to the three-dimensional Coulomb-like systems. The “Kähler oscillator” is a distinguished system with respect to supersymmetrisation as well. Its preliminary studies were presented in [6, 3]. In this paper we present the exact solution of the quantum oscillator on arbitrary-dimensional I CP, LN and related cones in the presence of a constant magnetic field. The study of such systems is not merely of academic interest. It is also relevant to the higher-dimensional quantum Hall effect. This theory has been formulated initially on the four-dimensional sphere [7] and further included, as a particular case, in the theory of the quantum Hall effect on complex projective spaces [8] (see, also [9]). The latter theory is based on the quantum mechanics on I CP in a constant magnetic field. Our basic observation is that the inclusion of the constant magnetic field does not break any existing hidden symmetries of the I CP-oscillator and, consequently, its superintegrability and/or exact solvability are preserved. To be more concrete, let us consider first the (classical) oscillator on IR = I C. It is described by the symplectic structure Ω0 = dπa ∧ dz + dπ̄a ∧ dz̄ (3) and the Hamiltonian H = ππ̄ + ωzz̄. (4)