Estimates of the gaps between consecutive eigenvalues of Laplacian

Daguang Chen, Tao Zheng and Hongcang Yang

Vol. 282 (2016), No. 2, 293–311

DOI: 10.2140/pjm.2016.282.293

Abstract

For the eigenvalue problem of the Dirichlet Laplacian on a bounded domain in Euclidean space ${ℝ}^n$, we obtain estimates for the upper bounds of the gaps between consecutive eigenvalues which are the best possible in terms of the orders of the eigenvalues. Therefore, it is reasonable to conjecture that this type of estimate also holds for the eigenvalue problem on a Riemannian manifold. We give some particular examples.

Keywords

Laplacian, consecutive eigenvalues, test function, Riemannian manifold, hyperbolic space

Mathematical Subject Classification 2010

Primary: 35P15, 58C40

Secondary: 58J50

Milestones

Received: 17 November 2014

Accepted: 11 October 2015

Published: 3 March 2016