Some Eigenvalue Inequalities for the Schrödinger Operator on Integer Lattices

TL;DR

This work studies eigenvalue inequalities for the Schrödinger operator on finite subsets of the integer lattice $\mathbb{Z}^n$, bridging discrete graph settings with classical continuous results. By formulating the eigenproblem $-\Delta u + V u = \lambda \rho u$ with $V\ge 0$ and $\rho>0$, the authors develop a graph-based framework and derive Yang's first- and second-type inequalities, along with HP and PPW-type bounds, for the spectrum of $\widetilde{H}=(-\Delta+V)/\rho$. The results generalize the discrete Laplacian inequalities to Schrödinger operators on graphs and recover the unweighted lattice inequalities when $\rho\equiv 1$. The approach relies on the carré du champ $\Gamma$, Green's formula, and Rayleigh-quotient techniques applied to carefully constructed trial functions along lattice coordinates.

Abstract

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.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 3.1
• proof
• proof : Proof of Theorem 1.1
• proof : Proof of Theorem 1.2
• proof : Proof of Theorem 1.3
• proof : Proof of Theorem 1.4