Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi-classical Heat Kernel Asymptotics and Morse Inequalities

Eric Jian-Ting Chen

Abstract

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.

Semi-classical Heat Kernel Asymptotics and Morse Inequalities

Abstract

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 , near the critical point of a given Morse function, as . It is shown that this family is approximately close to the heat kernel with respect to a system of the harmonic oscillators attached to . 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.
Paper Structure (17 sections, 234 equations)

This paper contains 17 sections, 234 equations.

Table of Contents

  1. Introduction
  2. Statements of the Main Results
  3. Preliminaries
  4. Notations and Terminologies
  5. Witten Laplacians and Heat Kernels
  6. Analytic Tools
  7. Scaled Heat Kernel Asymptotics
  8. Localization
  9. Scaling Technique
  10. Locally Uniform Bound for Scaled Heat Kernels
  11. Proof of Theorem 1.1
  12. Heat Kernel Asymptotics away from Critical Points
  13. Proof of Theorem 1.2
  14. Proof of Theorem 1.3
  15. Morse Inequalities
  16. ...and 2 more sections

Theorems & Definitions (22)

  • proof
  • proof
  • proof : Proof of Theorem \ref{['Thm: Locally Flat Metric']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['Main Thm 1: locally uniform boundedness of s.h.']}
  • proof
  • ...and 12 more