A proof of the nodal structure of the wave functions of supersymmetric partner potentials

TL;DR

This paper proves the nodal structure of wave functions in supersymmetric quantum mechanics using the quantum Hamilton-Jacobi formalism, confirming Gozzi's criterion for both unbroken and broken SUSY cases, including isospectral deformations.

Contribution

It provides a rigorous proof of the nodal difference in eigenstates of supersymmetric partner potentials, extending Gozzi's criterion to cases with isospectral deformation.

Findings

Confirmed that the nodal difference is one for unbroken SUSY

Confirmed that the nodal difference is zero for broken SUSY

Extended proof to include isospectral deformations

Abstract

Quantum Hamilton-Jacobi formalism is used to give a proof for Gozzi's criterion that for eigenstates of the supersymmetric partners, corresponding to same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We show that this proof is also applicable to the case, where isospectral deformation is involved.