Supersymmetry aspects of the Dirac equation in one dimension with a Lorentz scalar potential.

Abstract

The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\odinger Hamiltonians ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$. The ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ share the same energy spectrum and scattering phases. The shared spectrum includes the lowest states unless the Dirac equation allows a zero mode (a zero-energy bound state). This situation is unlike the common examples of supersymmetric quantum mechanics. The Dirac equation admits a zero mode only if the scalar potential has certain ``topology.'' Various such features are illustrated through explicit examples. In particular, the phase equivalence between ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ is exploited to construct transparent potentials for the Dirac equation in one dimension.