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 Schr\"odinger 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.