On supersymmetric Dirac delta interactions

In this paper we constructN = 2 supersymmetric quantum mechanics over several configurations of Dirac delta potentials from one single delta to a Dirac “comb ”. 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 δs is the singular potential formed by a Dirac δ plus a step at the same point. In this δ/θ 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 energy positive depending on the relation between the Dirac well strength and the height of the step potential. We find an scenario of either unbroken supersymmetry with Witten index one or supersymmetry breaking when there is one “bosonic”and one “fermionic”ground state such that the Witten index is zero. We explain next the different structure of the scattering waves produced by three δ/θ potentials with respect to the the eigenfunctions arising in the non-SUSY case. In particular, much more bound states paired by supersymmetry exist within the supersymmetric framework as compared with the non-SUSY problem. An infinite array of equally spaced δ-interactions of the same strength but alternatively attractive and repulsive are susceptible of being promoted to a N = 2 supersymmetric system. The Bloch’s theorem for wave functions in periodic potentials prompts a band spectrum also paired by supersymmetry. Self-isospectrality between the two partner Hamiltonians is thus found. Zero energy ground states are the non propagating band lower edges, which exist in the spectra of both the two diagonal operators forming the SUSY Hamiltonian. We find that for the SUSY Dirac “comb”the naif Witten index is zero but supersymmetry is unbroken. 1