R. D. Mota

su(1, 1) algebraic approach of the Dirac equation with Coulomb-type scalar and vector potentials in D+1 dimensions

We study the Dirac equation with Coulomb-type vector and scalar potentials in D+1 dimensions from an su(1, 1) algebraic approach. The generators of this algebra are constructed by using the Schrodinger factorization. The theory of unitary representations for the su(1, 1) Lie algebra allows us to obtain the energy spectrum and the supersymmetric ground state. For the cases where there exists either scalar or vector potential our results are reduced to those obtained by analytical techniques.

An su(1, 1) algebraic approach for the relativistic Kepler–Coulomb problem

We apply the Schr?dinger factorization method to the radial second-order equation for the relativistic Kepler?Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1, 1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.

The SU(1, 1) Perelomov number coherent states and the non-degenerate parametric amplifier

We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.

Creation and annihilation operators, symmetry and supersymmetry of the 3D isotropic harmonic oscillator

We show that the supersymmetric radial ladder operators of the three-dimensional isotropic harmonic oscillator are contained in the spherical components of the creation and annihilation operators of the system. Also, we show that the constants of motion of the problem, written in terms of these spherical components, lead us to second-order radial operators. Further, we show that these operators change the orbital angular momentum quantum number by two units and are equal to those obtained by the Infeld–Hull factorization method.

Laplace-Runge-Lenz vector, ladder operators and supersymmetry

We show that the Laplace-Runge-Lenz vector (LRLV) generates the Infeld-Hull radial factorization and the pair of isospectral Hamiltonians for the non-relativistic Kepler-Coulomb quantum problem. To do this we only use the LRLV and arguments of general soundness. Finally, the well known restrictions on the orbital angular momentum and the principal quantum numbers are rederived from the corresponding ladder operators.

The su(1,1) dynamical algebra from the Schrödinger ladder operators for N-dimensional systems: hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator

We apply the Schrodinger factorization to construct the ladder operators for the hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions. By generalizing these operators we show that the dynamical algebra for these problems is the su(1, 1) Lie algebra.

Constants of motion, ladder operators and supersymmetry of the two-dimensional isotropic harmonic oscillator

For the quantum two-dimensional isotropic harmonic oscillator we show that the Infeld–Hull radial operators, as well as those of the supersymmetric approach for the radial equation, are contained in the constants of motion of the problem.

SU(1,1) solution for the Dunkl oscillator in two dimensions and its coherent states

Abstract.We study the Dunkl oscillator in two dimensions by the su(1,1) algebraic method. We apply the Schrödinger factorization to the radial Hamiltonian of the Dunkl oscillator to find the su(1,1) Lie algebra generators. The energy spectrum is found by using the theory of unitary irreducible representations. By solving analytically the Schrödinger equation, we construct the Sturmian basis for the unitary irreducible representations of the su(1,1) Lie algebra. We construct the SU(1,1) Perelomov radial coherent states for this problem and compute their time evolution.

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödingers uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K0 of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.

ON THE SUPERSYMMETRY OF THE DIRAC–KEPLER PROBLEM PLUS A COULOMB-TYPE SCALAR POTENTIAL IN (D+1) DIMENSIONS AND THE GENERALIZED LIPPMANN–JOHNSON OPERATOR

We will study the Dirac–Kepler problem plus a Coulomb-type scalar potential by generalizing the Lippmann–Johnson operator to D spatial dimensions. From this operator, we construct the supersymmetric generators to obtain the energy spectrum for discrete excited eigenstates and the radial spinor for the SUSY ground state.