On two-dimensional superpotentials: from classical Hamilton-Jacobi theory to 2D supersymmetric quantum mechanics

TL;DR

This paper explores the relationship between classical Hamilton-Jacobi theory and 2D supersymmetric quantum mechanics, revealing how superpotentials relate to separability and presenting new supersymmetric extensions of classical systems.

Contribution

It demonstrates the connection between classical superpotentials and Hamilton-Jacobi theory in 2D, and introduces novel supersymmetric models for classical systems like the harmonic oscillator.

Findings

Superpotentials correspond to Hamilton characteristic functions in classical mechanics.

Multiple superpotentials can govern Hamilton-Jacobi separable systems with more than one degree of freedom.

The planar anisotropic harmonic oscillator admits two distinct supersymmetric extensions.

Abstract

Superpotentials in ${\cal N}=2$ supersymmetric classical mechanics are no more than the Hamilton characteristic function of the Hamilton-Jacobi theory for the associated purely bosonic dynamical system. Modulo a global sign, there are several superpotentials ruling Hamilton-Jacobi separable supersymmetric systems, with a number of degrees of freedom greater than one. Here, we explore how supersymmetry and separability are entangled in the quantum version of this kind of system. We also show that the planar anisotropic harmonic oscillator and the two-Newtonian centers of force problem admit two non-equivalent supersymmetric extensions with different ground states and Yukawa couplings.