Graded extension of so(2,1) Lie algebra and the search for exact solutions of the Dirac equation by point canonical transformations
- 2 April 2002
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review A
- Vol. 65 (4), 042109
- https://doi.org/10.1103/physreva.65.042109
Abstract
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.Keywords
Other Versions
This publication has 40 references indexed in Scilit:
- Relativistic extension of shape-invariant potentialsJournal of Physics A: General Physics, 2001
- Solvable potentials associated with su(1,1) algebras: a systematic studyJournal of Physics A: General Physics, 1994
- Mapping of shape invariant potentials under point canonical transformationsJournal of Physics A: General Physics, 1992
- Exact solutions for nonpolynomial potentials in N-space dimensions using a factorization method and supersymmetryJournal of Mathematical Physics, 1991
- Supersymmetric quantum mechanics of one-dimensional systemsJournal of Physics A: General Physics, 1985
- Group theory approach to scatteringAnnals of Physics, 1983
- Aspects of supersymmetric quantum mechanicsAnnals of Physics, 1983
- Potential Scattering, Transfer Matrix, and Group TheoryPhysical Review Letters, 1983
- On Dis and RacsPhysics Letters B, 1980
- General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functionsTheoretical and Mathematical Physics, 1979