It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{mathbb R})$. As a consequence it is possible to find
a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.
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 rqrq. We show in detail how the building of supersymmetry on potentials w
ith 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 bosonicrqrq and one fermionicrqrq 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...
