Study on the general adiabatic expansion theory for heat kernels and its some applications

阐明了一般绝热展开理论在研究与热核有关的一些问题中的有效性和威力。特别地，基于这一理论，我们得到了与黎曼拉普拉斯算子相关的热核的每个导数的渐近展开系数的公式。为了将高达任意阶的系数显式地描述为由曲率构造的万能多项式，我们只需要在公式中添加微积分的基本知识。(传统的方法需要各种深厚的知识，所以只研究了几个系数。)我们还发现了绝热膨胀与物理学家经常提到的某些反常现象之间的关系，并引入了一个有趣的反常公式。

We clarified the usefulness and powerfulness of the general adiabatic expansion theory in studying some subjects related to the heat kernels. In particular, based on the theory we obtained a formula for the coefficients of the asymptotic expansion of every derivative of the heat kernel associated to the Riemannian Laplacian. In order to describe the coefficients explicitly up to an arbitrarily high order as universal polynomials built from the curvature, we need only a basic knowledge of calculus added to the formula. (The conventional method requires various kinds of profound knowledge so that only a few coefficients were investigated.) We found also a relation between the adiabatic expansion and certain anomalies which are often referred to by physicists, and introduced an interesting formula for the anomalies.