Constant magnetic field and 2d non-commutative inverted oscillator

We consider a two-dimensional non-commutative inverted oscillator in the presence of a constant magnetic field, coupled to the system in a “symplectic” and “Poisson” way. We show that it has a discrete energy spectrum for some value of the magnetic field. PACS number: 03.65.-w Introduction Non-commutative quantum field theories have been studied intensively during the last several years, owing to their relationship with M-theory compactifications [1], string theory in nontrivial backgrounds [2] and quantum Hall effect [3] (see e.g. [4] for a recent review). At low energies the one-particle sectors become relevant, which prompted an interest in the study of non-commutative quantum mechanics (NCQM) [5] [19] (for some earlier studies of NCQM see [20] [22]). Most of the attention was focused on quantum mechanics on twoand three-dimensional noncommutative spaces. Two-dimensional NCQM in the presence of a constant magnetic field was considered on a plane [8, 11], torus [9], sphere [8] and pseudosphere (Lobachewski plane, or AdS2) [16, 19]. NCQM on a plane has a critical point, specified by the vanishing of the dimensionless parameter κ = 1−Bθ , (1) where the system becomes effectively one-dimensional [8, 11]. Away from the critical point, the rotational properties of the model become qualitatively dependent on the sign of κ: for κ > 0 the system admits an infinite number of states with a given value of the angular momentum, while for κ < 0 the number of such states is finite [11]. From NCQM on a (pseudo)sphere originate, in some sense, the “phases” of planar NCQM [13]: the “monopole number” is defined, in such phases, in a different way. In the planar limit the NCQM on (pseudo)sphere results in the “non-conventional”, or the so-called “exotic” NCQM [10], where the magnetic field is introduced via “minimal”, or symplectic coupling. Notice that the only two-dimensional NCQM with a non-zero potential term that has been solved explicitly corresponds to a noncommutative circular oscillator in the presence of a magnetic field. Although this system has been solved for both conventional and exotic coupling of the magnetic field, its rotational properties have been analyzed only for a conventional coupling of the magnetic field. The present interest in NCQM was initiated by Chaichian, Sheikh-Jabbari and Tureanu [6], who calculated the corrections to the hydrogen atom spectrum, which arise in non-commutative QED. Later on, the three-dimensional noncommutative oscillator with a conventional coupling of the magnetic field has also been considered and explicitly solved [23]. Hence, the recent studies of NCQM concentrated mainly on the systems with either an attractive potential, or without potential. On the other hand, Ho and Kao [18], accurately considering the multi-particle non-relativistic limit of the noncommutative field theory, found that particles of opposite charges have opposite non-commutativity. Hence, in the center-of-mass frame, the “particle-antiparticle” system has no noncommutative correction, in contrast to the system of two identical particles. Therefore, in the context of a two-particle interpretation, NCQM with a repulsive potential becomes especially important. We recall that, at the present time, the only example of an exactly solvable NCQM system with a nonzero potential term is the oscillator, which adds on to the distinguished role played by the harmonic 1In [18] it was claimed, that “since proton and electron have opposite charges, the hydrogen atom has no noncommutative corrections, in spite of the statement of [6]”. However, the authors of the latter paper argued in their reply [24] that this prediction does not work in the case of the hydrogen atom, owing to the composite nature of the proton.