在這篇論文中，我們引入了在半經典分析中的伸縮技巧和定義伸縮熱核。我們透過研究伸縮熱核的漸近行為去捕捉摩斯函數的臨界點。除此之外，我們研究在臨界點外面的熱核漸近行為。作為一個應用，這些漸近行為的結果給出了一個對於摩斯不等式的全新解析證明。

In this thesis, we introduce the scaling technique in semi-classical analysis
and define the scaled heat kernels. We capture critical points of a Morse function by
studying the asymptotic behaviors of the scaled heat kernels. We also study the asymptotic
behaviors of heat kernels outside critical points. As an application, these asymptotic
behavior results give a new analytic proof of the Morse inequalities.

Acknowledgements-------v

Chapter 1. Introduction-------1

Chapter 2. Preliminaries------7
2.1. Notations and Terminologies-------7
2.2. Hodge Laplacians-------8
2.3. Morse Functions-------10

Chapter 3. Witten Laplacians-------13
3.1. Witten Deformation-------13
3.2. Witten Laplacians-------14

Chapter 4. Semi-classical Heat Kernels for Witten Laplacians-------19
4.1. Heat Kernels-------19
4.2. Locally Flat Metric near Critical Point-------21
4.3. Scaling Technique-------24

Chapter 5. Scaled Heat Kernel Asymptotics-------29
5.1. Sobolev Space Theory-------29
5.2. Mapping Property of Scaled Heat Operators-------33
5.3. Proof of Theorem 4.5-------36
5.4. Asymptotic Behaviors of Scaled Heat Kernels-------39
5.A. Appendix: Proof of (5.4.6)-------43

Chapter 6. Heat Kernel Asymptotics outside Critical Points-------45
6.1. Bochner Method-------45
6.2. Asymptotic Behaviors of Heat Kernels outside Critical Points-------52
6.3. Application to Morse inequalities-------56

Bibliography-------59

[1] C. Y. Hsiao, W. Zhu, Heat Kernel Asymptotics for Kohn Laplacians on CR Manifolds,
arXiv:2106.09268.
[2] J. Lee, Introduction to Riemannian Manifolds, 2nd edition. Springer, 2018.
[3] G. Marinescu, The Laplace Operator on High Tensor Powers of Line Bundles. Preprint.
[4] H. McKean, I. Singer, Curvature and the eigenvalue of the Laplacian, J. Differential Geometry 1 (1967), 43-69.
[5] J. Milnor, Morse Theory. Princeton University Press, Princeton, 1963.
[6] L. Nicolaescu, An Invitation to Morse Theory, 2nd edition. Springer, 2011.
[7] F. Warner, Foundations of Differentiable Manifolds and Lie Groups. Springer, 1983.
[8] E. Witten, Supersymmetry and Morse Theory, J. Differential Geometry 17 (1982), 661-692.