A new simple class of superpotentials in SUSY Quantum Mechanics

TL;DR

This paper introduces a new class of superpotentials in SUSY quantum mechanics, analyzing specific cases to find exact and approximate solutions, and comparing these with existing results.

Contribution

It presents a novel class of superpotentials $W(x)=g\epsilon(x) x^{2n}$ and explores their implications for solvable quantum systems, including exact and variational solutions.

Findings

Exact ground state for $V_{-}$ potential.

Variational approximation for excited states.

Comparison with existing literature results.

Abstract

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.