One-dimensional scattering of fermions on $delta$-impurities

We study the spectrum of the 1D Dirac Hamiltonian encompassing the bound and scattering states of a fermion distorted by a static background built from $delta$-function potentials. We distinguish between mass-spike and electrostatic $delta$-potentials. Differences in the spectra arising depending on the type of $delta$-potential studied are thoroughly explored.