Upper bounds for eigenvalues of the discrete and continuous Laplace operators

Fan RK Chung A Grigor'Yan Shing-Tung Yau

Convex and Discrete Geometry mathscidoc:1912.43525

advances in mathematics, 117, (2), 165-178, 1996.2
In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics. We develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel. This approach is perhaps much more general than its realization here. Basically, we start with the following entries:(1) an underlying space M with a finite measure+;(2) a well-defined Laplace operator 2 on functions on M so that 2 is a self-adjoint operator in L2 (M,+) with a discrete spectrum;(3) if M has a boundary, then the boundary condition should be chosen so that it does not disrupt self-adjointness of 2 and is of dissipative nature;(4) a distance function dist (x, y) on M so that|{dist| 1 for an appropriate notion of gradient.
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@inproceedings{fan1996upper,
  title={Upper bounds for eigenvalues of the discrete and continuous Laplace operators},
  author={Fan RK Chung, A Grigor'Yan, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203819297966089},
  booktitle={advances in mathematics},
  volume={117},
  number={2},
  pages={165-178},
  year={1996},
}
Fan RK Chung, A Grigor'Yan, and Shing-Tung Yau. Upper bounds for eigenvalues of the discrete and continuous Laplace operators. 1996. Vol. 117. In advances in mathematics. pp.165-178. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203819297966089.
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