A new simple class of superpotentials in SUSY Quantum Mechanics

In this work we introduce the class of quantum mechanics superpotentials $W(x)=g\epsilon(x) x^{2n}$ 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 potentials $V_{\pm}(x)=g^{2}x^{4}\pm2g|x|$. For $V_{-}$ 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 state in the literature.