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On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space. (English) Zbl 1358.81133
This paper treats non-relativistic two-dimensional rotationally invariant Hamiltonians on the non-commutative phase space, where the Hermitian operators representing coordinates \(\hat{X}_i\) and momenta \(\hat{P}_i\), \(i=1,2\), satisfy the commutation relations \[ [ \hat{X}_i, \hat{X}_j ] = \imath \theta \epsilon_{ij}, \quad [ \hat{X}_i, \hat{P}_j ] = \imath \hslash \delta_{ij}, \quad [ \hat{P}_i, \hat{P}_j ] = \imath \kappa \epsilon_{ij}, \quad \] with the real non-commutative parameters \(\theta \geq 0\) and \(\kappa\). The authors show that as a consequence of this commutator algebra, the rotationally invariant Hermitian quadratic forms in \(\hat{X}_i\), \(\hat{P}_i\), \(i=1,2\) generate a non-abelian tree-dimensional Lie algebra corresponding to \(sl(2,\mathbb R)\) or \(su(2)\) according to \(\kappa \theta < \hslash^2\) or \(\kappa \theta > \hslash^2\). Hence, constructed is the rotation generator \(\hat{L}\) in both the coordinate and momenta planes, where a dimensional reduction takes place as indicated in [V. P. Nair and A. P. Polychronakos, Phys. Lett., B 505, No. 1–4, 267–274 (2001; Zbl 0977.81046)]. Note that the generator \(\hat{L}\) is singular at the critical value \(\kappa \theta=\hslash^2\). In addition, they also construct operators which perform the discrete transformations of time-reversal and parity in each region. Then the most general rotationally invariant non-relativistic Hamiltonian can be expressed as a function of the generators of these Lie algebras and \(\hat{L}\) itself, and then its characteristic subspaces prove to be contained in unitary irreducible representation of the groups \(\mathrm{SL}(2,\mathbb R)\otimes\mathrm{SO}(2)\) or \(\mathrm{SU}(2)\otimes\mathrm{SO}(2)\), according to the region. Moreover, the authors show that the quadratic Casimir operator of the representations is related with the eigenvalue of \(\hat{L}\), namely, the relation which selects the physically sensible representations of these direct products. In this framework, they consider the simple examples of the extensions to these non-commutative phase space of the Hamiltonians of the isotropic oscillator and the Landau model, reproducing from this perspective their well known spectra. Finally the case of cylindrical well potentials on the plane is considered as well. They also solve the general case for the different parameters regions. In particular, for the case of infinite wells, they succeed in getting the eigenvalues corresponding to each irreducible representation as the zeros of given polynomials.

MSC:
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R60 Noncommutative geometry in quantum theory
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