Bound states of 2+1 fermionic trimers on lattice at strong couplings

TL;DR

This paper studies the existence of bound states of three-fermion systems on a lattice at strong coupling, revealing how mass ratios and quasi-momentum influence the spectral properties and emergence of bound states.

Contribution

It provides explicit thresholds for the mass ratio determining the presence of bound states in a three-fermion lattice model at strong coupling.

Findings

Bound states depend on mass ratio and quasi-momentum.

Thresholds for bound state existence are explicitly calculated for zero quasi-momentum.

Bound states can appear within the spectral gap at large coupling for certain mass ratios.

Abstract

In this paper, we investigate the bound states of $2+1$ fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schr\"odinger operator $H_{\gamma,\lambda}(K),$ focusing on energies below the continuum and within its gap. Depending on the quasi-momentum $K,$ we show that if the mass ratio $\gamma>0$ between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling $\lambda>0.$ Conversely, if $\gamma$ exceeds this threshold, $H_{\gamma,\lambda}(K)$ admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as $\lambda\to+\infty.$ For $K=0,$ the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.