Around supersymmetry for semiclassical second order differential operators

TL;DR

This paper investigates conditions under which a supersymmetric structure for semiclassical second order differential operators exhibits favorable estimates, enhancing understanding of their mathematical properties.

Contribution

It provides a sufficient condition on the coefficients of the operator ensuring the supersymmetric matrix structure has good semiclassical estimates.

Findings

Established a new sufficient condition for supersymmetric matrix estimates.

Improved understanding of the coefficient conditions for supersymmetry.

Enhanced mathematical tools for analyzing semiclassical differential operators.

Abstract

Let $P(h),h\in]0,1]$ be a semiclassical scalar differential operator of order $2$. The existence of a supersymmetric structure given by a matrix $G(x;h)$ was exhibited in \cite{HeHiSj13} under rather general assumptions. In this note we give a sufficient condition on its coefficient so that the matrix $G(x;h)$ enjoys some nice estimates with respect to the semiclassical parameter.