Resonant states in an attractive one dimensional cusp potential

We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit-Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.