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

The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schrödinger Hamiltonians H1 and H2. The H1 and H2 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 H1 and H2 is exploited to construct transparent potentials for the Dirac equation in one dimension.