Quasi-exact solvability of Dirac–Pauli equation and generalized Dirac oscillators

Abstract

In this paper we demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasiexactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the sl(2) symmetry are discussed separately in spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.

Simultaneous ordinary and type A N-fold supersymmetries in Schrödinger, Pauli, and Dirac equations

“…For the excited states, we cannot proceed the investigation as in the trigonometric case due to the lack of shape invariance. However, we can check by using the method of Bethe ansatz equations [6,7,12] that no elements of V ± N can be normalizable physical eigenstates except for the ground state ψ…”

“…In view of the fact that so far all the N -fold SUSY potentials are only one-dimensional, it is natural that one should look for physical models which are effectively one-dimensional. Experience gained in the work in [6,7] suggests that Pauli and Dirac equations are good candidates to start with. In this respect we note that the authors of [24] found that if the Pauli equation is generalized such that the gyromagnetic ratio g = 2 of the electron is changed to some unphysical values g = 2N (N ≥ 2), then for certain forms of magnetic fields, the generalized Pauli equation could possess type A N -fold SUSY.…”

“…It is based on the fact that, for some electromagnetic field configurations, Pauli and Dirac equations possess intrinsic SUSY structure, and different components of a wave function are related by some supercharges [4,5]. More recently, the method of SUSY has been employed to construct quasi-exactly solvable potentials in the Pauli and Dirac equations [6,7]. For quasi-exactly solvable systems, it is only possible to determine algebraically a part of the spectrum but not the whole spectrum [8][9][10][11][12].…”

“…Therefore, we would like to extend the realistic systems in [6,7] to include N -fold SUSY. Since the Pauli and Dirac equations considered there all possess ordinary SUSY, it is therefore natural that we look for an ordinary supersymmetric system which has N -fold supersymmetry as well, as a starting point of the aforementioned purpose.…”

Quasi-Exact Solvability of Planar Dirac Electron in Coulomb and Magnetic Fields

“…Later, several physical QES models were discovered, which include the two-dimensional Schrödinger [10], the Klein-Gordon [11], and the Dirac equation [12,13] of an electron moving in an attractive/repulsive Coulomb field and a homogeneous magnetic field. More recently, the Pauli and the Dirac equation minimally coupled to magnetic fields [14], and Dirac equation of neutral particles with non-minimal electromagnetic couplings [15] were also shown to be QES.…”