Graded extension of so(2,1) Lie algebra and the search for exact solutions of the 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 Mörse 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-Mörse potentials. This class is, in fact, the relativistic extension of the oscillator class. An extended point canonical transformation, which is compatible with the relativistic problem, is formulated. It maps all of these relativistic potentials into the Dirac-Oscillator potential.