Some Eigenvalue Inequalities for the Schrödinger Operator on Integer Lattices

In this paper, we establish analogues of the Payne-Pólya-Weinberger, Hile-Protter, and Yang eigenvalue inequalities for the Schrödinger 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 Schrödinger operators with nonnegative potentials and weighted eigenvalue problems.