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

Abstract

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.