In this work we introduce the class of quantum mechanics superpotentials W (x) = gε(x)x 2n and study in details 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 result in the potentialsFor V − we present the exact ground state solution and study the excited states by a variational technic. Starting from the ground state of V − and using logarithmic perturbation theory we study the ground states of V + and also of V (x) = g 2 x 4 and compare the result got by this new way with other results for this last potential in the literature. I. INTRODUCTION Supersymmetric quantum mechanics (SUSY QM) was first introduced by E. Witten [1][2], as a simplified model (a 0 + 1 dimensional field theory) to study the possibility of SUSY breaking. Soon it became a research branch in itself, a way of getting new solutions to problems in quantum mechanics [3] [4] [5] [6]. Of particular interest, to our work below, we must cite the many papers in the literature [7] [8] [9] [10] [11] [12] [13] [14] devoted to the development of technics for treating the anharmonic oscillator V (x) = ω 2 x 2 + g 2 x 4 , and other related potentials, which in general do not have exact solutions.In this work we present a new simple class of superpotentials in SUSY QM, in the form W (x) = gε(x)x 2n with n = 0, 1, 2, . . .. The first example of this class, i.e., the case n = 0, *

We discuss the scattering of relativistic spin zero particles by an infinitely long and arbitrarily thin solenoid. The exact solution of the first-quantized problem can be obtained as a mimic of the nonrelativistic case, either in the original Aharonov-Bohm way or by using the Berry's magnetization scheme. The perturbative treatment is developed in the Feshbach-Villars two-component formalism for the Klein-Gordon equation and it is shown that it also requires renormalization as in the Schrödinger counterpart. The results are compared with those of the field theoretical approach which corresponds to the two-body sector of the scalar Chern-Simons theory.