Semi-classical Heat Kernel Asymptotics and Morse Inequalities

In this paper, we study the asymptotic behavior of the heat kernel with respect to the Witten Laplacian. We introduce the localization and the scaling technique in semi-classical analysis, and study the semi-classical asymptotic behavior of the family of the heat kernel, indexed by $k$, near the critical point $p$ of a given Morse function, as $k\to \infty$. It is shown that this family is approximately close to the heat kernel with respect to a system of the harmonic oscillators attached to $p$. We also furnish some asymptotic results regarding heat kernels away from the critical points. These heat kernel asymptotic results lead to a novel proof of the Morse inequalities.