Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet Laplace eigenvalues on integer lattices

TL;DR

This paper establishes new inequalities for Dirichlet Laplace eigenvalues on integer lattices, extending classical results to discrete settings and addressing a question by Chung and Oden.

Contribution

It introduces analogues of classical eigenvalue inequalities for discrete Laplacians on integer lattices, expanding the theoretical understanding of spectral properties in discrete spaces.

Findings

Derived inequalities analogous to Payne-Polya-Weinberger, Hile-Protter, and Yang for discrete Laplacians.

Partially answered a question posed by Chung and Oden regarding eigenvalue bounds.

Extended classical spectral inequalities to the setting of integer lattices.

Abstract

In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.