No zero energy states for the supersymmetric x^2y^2 potential

TL;DR

This paper proves that a specific supersymmetric matrix-valued differential operator with a x^2 y^2 potential has no zero-energy states, indicating the absence of zero modes in this system.

Contribution

It demonstrates the non-existence of zero modes for a particular supersymmetric operator with a quadratic potential, advancing understanding of supersymmetric spectral properties.

Findings

No zero modes for the operator H

Zero energy states imply trivial solutions

Supports theoretical predictions about supersymmetric operators

Abstract

We show that the positive supersymmetric matrix-valued differential operator H={p_x}^2 + {p_y}^2 + x^2y^2 + x\sigma_3 + y\sigma_1 has no zero modes, i.e., H \psi = 0 implies \psi =0.