  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.
