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

2007/03/28 by Wu-Sheng Dai, Dai, Wu-Sheng, Mi Xie +1
Mathematics · #FOS: Mathematics #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.math/0703847

5 pages, no figure

arxiv created 2008/02/17 · arxiv updated 2009/12/01

Abstract

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 λ, and the heat kernel K(t), defined by K(t)=∑_ne^-lambda_nt. Moreover, we also give an asymptotic formula for N(λ) and discuss when lambda → ∞ in what cases N(lambda)=K(1/lambda).

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