Some Eigenvalue Inequalities for the Schr\"odinger Operator on Integer Lattices

TL;DR

This paper extends classical eigenvalue inequalities to Schr"odinger operators on integer lattices, providing bounds for eigenvalues in discrete settings with potentials.

Contribution

It introduces discrete analogues of well-known inequalities for Schr"odinger operators on finite subsets of integer lattices, generalizing previous results for the Laplacian.

Findings

Established eigenvalue inequalities for Schr"odinger operators on integer lattices.

Extended classical inequalities to discrete Schr"odinger operators with potentials.

Provided bounds applicable to weighted eigenvalue problems.

Abstract

In this paper, we establish analogues of the Payne-P\'olya-Weinberger, Hile-Protter, and Yang eigenvalue inequalities for the Schr\"odinger operator on arbitrary finite subsets of the integer lattice $\mathbb{Z}^n$. The results extend known inequalities for the discrete Laplacian to a more general class of Schr\"odinger operators with nonnegative potentials and weighted eigenvalue problems.