Semi-classical trace formulas and heat expansions

TL;DR

This paper demonstrates how to derive a semi-classical trace formula for Schrödinger operators with magnetic fields using heat expansion techniques developed by geometers in the 1970s.

Contribution

It shows how to recover the second term of the semi-classical trace formula using heat expansion methods, connecting geometric analysis with quantum spectral theory.

Findings

Derived the second term of the semi-classical trace formula

Connected heat expansion methods with semi-classical spectral analysis

Provided a geometric perspective on quantum trace formulas

Abstract

in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schr\"odinger operator with magnetic field. We show how to recover their formula by using the methods developped by the geometers in the seventies for the heat expansions.