A new simple class of superpotentials in SUSY quantum mechanics

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Abstract: In this work we introduce the class of quantum mechanics superpotentials

W (x) = g e p s i l o n (x) x^{2 n}

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_{p m} (x) = g^{2} x^{4} p m 2 g | 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.