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

On supersymmetric Dirac delta interactions