4 research outputs found
Graded extension of SO(2,1) Lie algebra and the search for exact solutions of Dirac equation by point canonical transformations
SO(2,1) is the symmetry algebra for a class of three-parameter problems that
includes the oscillator, Coulomb and Morse potentials as well as other problems
at zero energy. All of the potentials in this class can be mapped into the
oscillator potential by point canonical transformations. We call this class the
"oscillator class". A nontrivial graded extension of SO(2,1) is defined and its
realization by two-dimensional matrices of differential operators acting in
spinor space is given. It turns out that this graded algebra is the
supersymmetry algebra for a class of relativistic potentials that includes the
Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in
fact, the relativistic extension of the oscillator class. A new point canonical
transformation, which is compatible with the relativistic problem, is
formulated. It maps all of these relativistic potentials into the
Dirac-Oscillator potential.Comment: Replaced with a more potrable PDF versio