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

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.