The relation between the counting function N(lambda) and the heat kernel K(t)

Details The relation between the counting function N(lambda) and the heat kernel K(t) The relation between the counting function N(lambda) and the heat kernel K(t)

2007-03-28

arxiv.org/abs/math/0703847v2

Mathematics

Spectral Theory

5 pages, no figure

Scientific paper

For a given spectrum {lambda_{n}} of the Laplace operator on a Riemannian manifold, in this paper, we present a relation between the counting function N(lambda), the number of eigenvalues (with multiplicity) smaller than \lambda, and the heat kernel K(t), defined by K(t)=\sum_{n}e^{-lambda_{n}t}. Moreover, we also give an asymptotic formula for N(\lambda) and discuss when lambda \to \infty in what cases N(lambda)=K(1/lambda).

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