Semi-classical Green kernel asymptotics for the Dirac operator

TL;DR

This paper derives semi-classical asymptotics for the Green kernel of the Dirac operator, revealing exponential decay modulated by an Agmon distance and providing explicit leading-term formulas.

Contribution

It establishes a detailed asymptotic expansion for the Dirac operator's Green kernel in arbitrary dimensions, including explicit leading-term formulas.

Findings

Green kernel exhibits exponential decay with Agmon distance

Asymptotic expansion in powers of semi-classical parameter

Explicit formula for the leading term

Abstract

We consider a semi-classical Dirac operator in arbitrary spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.