The number and location of two particle Schr\"odinger operators on a lattice

TL;DR

This paper analyzes the spectral properties of two-particle Schr"odinger operators on a 2D lattice, classifying parameter regions by the fixed number of eigenvalues and providing bounds on isolated eigenvalues.

Contribution

It introduces a classification of the parameter space for two-particle lattice Schr"odinger operators based on the number of eigenvalues, including bounds on isolated eigenvalues.

Findings

Partitioned the parameter space into regions with fixed eigenvalue counts.

Established bounds for the number of isolated eigenvalues.

Analyzed eigenvalues both below and above the essential spectrum.

Abstract

We study the Schr\"odinger operators ${H}_{\lambda\mu}(K)$ with $K\in\mathbb{T}^2$ being the fixed quasimomentum of a pair of particles, associated with a system of two arbitrary particles on a two-dimensional lattice $\mathbb{Z}^2$ with on-site and nearest-neighbor interactions of strengths $\lambda\in\mathbb{R}$ and $\mu\in\mathbb{R}$, respectively. We divide the $(\lambda,\mu)$-plane of parameters $\lambda$ and $\mu$ into connected components, such that in each component, the Schr\"{o}dinger operator $H_{\lambda\mu}(0)$ has a fixed number of eigenvalues. These eigenvalues are located both below the bottom of the essential spectrum and above its top. Additionally, we establish a sharp lower bound for the number of isolated eigenvalues of $H_{\lambda\mu}(K)$ within each connected component.