Quasi-exact Solvability of Dirac Equations

We present a general procedure for determining quasi-exact solvability of the Dirac and the Pauli equation with an underlying sl(2) symmetry. This procedure makes full use of the close connection between quasi-exactly solvable systems and supersymmetry. The Dirac-Pauli equation with spherical electric field is taken as an example to illustrate the procedure. 1. In this talk we present a general procedure for determining quasi-exact solvability of the Dirac and the Pauli equation with an underlying sl(2) symmetry. This procedure makes full use of the close connection between quasi-exactly solvable (QES) systems and supersymmetry (SUSY), or equivalently, the factorizability of the equation. Based on this procedure, we have demonstrated that the Pauli and the Dirac equation coupled minimally with a vector potential [1], neutral Dirac particles in external electric fields (which are equivalent to generalized Dirac oscillators) [2], and Dirac equation with a Lorentz scalar potential [3] are physical examples of QES systems. Here we only give the main ideas of the procedures, and refer the readers to [1, 2, 3] for details. ∗Based on talks presented at the 11th International Conference on Symmetry Methods in Physics (Jun 21-24, 2004, Prague) and at the XXIII International Conference on Differential Geometric Methods in Theoretical Physics (Aug 20-26, 2005, Nankai, Tianjin, China). 1 2. For all the cases cited above, one can reduce the corresponding multi-component equations to a set of one-variable equations possessing one-dimensional SUSY after separating the variables in a suitable coordinate system. Typically the set of equations takes the form