本征态的个数：计数函数与热核
The number of eigenstates: counting function and heat kernel
摘要：
本文的主要目的有两个：（1）揭示计数函数N（λ）（本征值小于给定值的本征态个数）与热核K（t）之间的关系，这在数学上是一个未解决的问题;（2）介绍一种计算N（λ）的方法，因为在首阶以上还没有有效的计算N（λ）的方法。本文提出了一种更适合于实际计算的新的N（lambda）表达式。本文构造了一个重正化过程，用以消除从K（t）的非一致收敛展开式得到N（lambda）时出现的发散。我们计算D维盒子，三维球和二维多连通不规则区域的N（lambda）。利用Gauss-Bonnet定理，将单连通热核推广到多连通热核，证明了Kac关于二维多连通热核的猜想。介绍了由N（λ）计算本征谱和态密度的方法。
英文摘要：
The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.
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