Discrete Spectrum of a Model Operator Related to Three-Particle Discrete Schr\"{o}dinger Operators

TL;DR

This paper analyzes the spectral properties of a three-particle lattice Schrödinger operator with nonlocal interactions, revealing infinitely many negative eigenvalues under certain resonance conditions and providing asymptotic eigenvalue counts.

Contribution

It establishes the existence of infinitely many negative eigenvalues and derives their asymptotic distribution for a specific three-particle lattice model with nonlocal potentials.

Findings

Infinitely many negative eigenvalues accumulate at zero.

Asymptotic behavior of eigenvalues near zero is characterized.

Resonance conditions lead to spectral accumulation.

Abstract

A model operator $H_\mu,$ $\mu>0$ associated to a system of three particles on the three-dimensional lattice $ \mathbb{Z}^3$ that interact via nonlocal pair potentials is considered. We study the case where the parameter function $w$ has a special form with the non degenerate minimum at the $n, n>1$ points of the six-dimensional torus $\mathbb{T}^6.$ If the associated Friedrichs model has a zero energy resonance, then we prove that the operator $H_\mu$ has infinitely many negative eigenvalues accumulating at zero and we obtain an asymptotics for the number of eigenvalues of $H_\mu$ lying below $z,$ $z<0$ as $z\to -0.$