On supersymmetric Dirac delta interactions

In this paper we construct $\mathcal{N}=2$ supersymmetric (SUSY) quantum mechanics over several configurations of Dirac-$\delta$ potentials from one single delta to a Dirac " comb \rq\rq. We show in detail how the building of supersymmetry on potentials with delta interactions placed in two or more points on the real line requires the inclusion of quasi-square wells. Therefore, the basic ingredient of a supersymmetric Hamiltonian containing two or more Dirac-$\delta$s is the singular potential formed by a Dirac-$\delta$ plus a step ($\theta$) at the same point. In this $\delta/\theta$ SUSY Hamiltonian there is only one singlet ground state of zero energy annihilated by the two supercharges or a doublet of ground states paired by supersymmetry of positive energy depending on the relation between the Dirac well strength and the height of the step potential. We find a scenario of either unbroken supersymmetry with Witten index one or supersymmetry breaking when there is one " bosonic\rq\rq and one " fermionic\rq\rq ground state such that the Witten index is zero. We explain next the different structure of the scattering waves produced by three $\delta/\theta$ potentials with respect to the eigenfunctions arising in the non-SUSY case. In particular, many more bound states paired by supersymmetry exist within the supersymmetric framework compared with the non-SUSY problem. An infinite array of equally spaced $\delta$-interactions of the same strength but alternatively attractive and repulsive are susceptible of being promoted to a ${\cal N}=2$ supersymmetric system...