Upper Bounds for Eigenvalues of the Discrete and Continuous Laplace Operators

Abstract

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 μ; 2◦ a well-defined Laplace operator ∆ on functions on M so that ∆ is a self-adjoint operator in L(M,μ) 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 ∆ and is of dissipative nature; 4◦ a distance function dist(x, y) on M so that |∇dist| ≤ 1 for an appropriate notion of gradient.