Estimates of the gaps between consecutive eigenvalues for a class of elliptic differential operators in divergence form on Riemannian manifolds

TL;DR

This paper provides upper bounds for gaps between consecutive eigenvalues of certain elliptic operators, including the Laplacian, on Riemannian manifolds, extending known results and applying to specific geometric contexts.

Contribution

It introduces new eigenvalue gap estimates for a broad class of elliptic operators in divergence form on Riemannian manifolds, generalizing previous results for the Laplacian.

Findings

Estimates match the best known bounds for the Laplacian.

Applicable to elliptic operators on pinched Cartan-Hadamard manifolds.

Provides bounds in limited Euclidean domains.

Abstract

In this work, we obtain estimates for the upper bound of gaps between consecutive eigenvalues for the eigenvalue problem of a class of second-order elliptic differential operators in divergent form, with Dirichlet boundary conditions, in a limited domain of n-dimensional Euclidean space. This class of operators includes the well-known Laplacian and the square Cheng-Yau operator. For the Laplacian case, our estimate coincides with that obtained by D. Chen, T. Zheng, and H. Yang, which is the best possible in terms of the order of the eigenvalues. For pinched Cartan-Hadamard manifolds the estimates were made in particular cases of this operator.