Supersymmetric structures for second order differential operators

TL;DR

This paper establishes conditions under which second order differential operators have supersymmetric structures and demonstrates the non-existence of such structures in a specific physical model involving oscillators and heat baths with different temperatures.

Contribution

It provides necessary and sufficient conditions for supersymmetric structures in second order PDEs and shows their non-existence in a physical oscillator model with temperature differences.

Findings

Conditions for supersymmetric structures are characterized.

Supersymmetric structures do not exist for certain oscillator models with different bath temperatures.

The results connect mathematical structures with physical heat bath models.

Abstract

Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For the operator coming from a chain of oscillators, coupled to two heat baths, we show the non-existence of a smooth supersymmetric structure, for a suitable interaction potential, provided that the temperatures of the baths are different.