Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites

TL;DR

This paper analyzes the number of bound states in a two-boson lattice system with interactions up to next-nearest neighbors, providing bounds and spectral properties of the associated Hamiltonian operators.

Contribution

It introduces an invariant subspace for the Hamiltonian and establishes bounds on the number of eigenvalues depending on interaction parameters.

Findings

Existence of an invariant subspace with at most two eigenvalues.

Bounds on the number of eigenvalues below and above the essential spectrum.

Dependence of eigenvalues on interaction magnitudes.

Abstract

We study the family $H_{\gamma \lambda \mu}(K)$, $K\in \mathbb{T}^2,$ of discrete Schr\"odinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimen\-sional lattice $\mathbb{Z}^2,$ interacting through on one site, nearest-neighbour sites and next-nearest-neighbour sites with interaction magnitudes $\gamma,\lambda$ and $\mu,$ respectively. We prove there existence an important invariant subspace of operator $H_{\gamma \lambda \mu}(0)$ such that the restriction of the operator $H_{\gamma \lambda \mu}(0)$ on this subspace has at most two eigenvalues lying both as below the essential spectrum as well as above it, depending on the interaction magnitude $\lambda,\mu\in \mathbb{R}$ (only). We also give a sharp lower bound for the number of eigenvalues of $H_{\gamma\lambda\mu}(K)$.