A new simple class of superpotentials in SUSY quantum mechanics
In this work, we introduce the class of quantum mechanics superpotentials W(x) = gε(x)x2n and study in detail the cases n = 0 and 1. The n = 0 superpotential is shown to lead to the known problem of two supersymmetrically related Dirac delta potentials (well and barrier). The n = 1 case results in the potentials V±(x) = g2x4 ± 2g|x|. For V−, we present the exact ground-state solution and study the excited states by a variational technique. Starting from the ground state of V− and using logarithmic perturbation theory, we study the ground states of V+ and also of V(x) = g2x4 and compare the result obtained in this new way with other results for this last potential in the literature.