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. After introducing the most general Dirac-$\delta$ potential for the Dirac equation we distinguish between \lq\lq mass-spike\rq\rq and \lq\lq electrostatic\rq\rq $\delta$-potentials. Differences in the spectra arising depending on the type of $\delta$-potential are studied in deep detail.