Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs

TL;DR

This paper introduces a discrete poly-Laplace operator on lattice subgraphs, providing bounds for eigenvalues and establishing inequalities between different order eigenvalues, extending classical spectral results.

Contribution

It extends classical eigenvalue bounds to the discrete poly-Laplace operator on lattice subgraphs, including new inequalities between eigenvalues of different orders.

Findings

Derived upper and lower bounds for eigenvalue sums.

Proved eigenvalue inequalities between different orders.

Extended classical spectral results to discrete poly-Laplace operators.

Abstract

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of lattice graph $\mathbb{Z}^{d}$ extending classical results of Li-Yau and Kr\"oger. Moreover, we prove that the Dirichlet $2l$-order poly-Laplace eigenvalues are at least as large as the squares of the Dirichlet $l$-order poly-Laplace eigenvalues.