# Efimov effect for a three-particle system with two identical fermions

## TL;DR

This paper rigorously proves the Efimov effect occurs in a three-particle quantum system with two identical fermions and a different particle when the mass ratio is below a critical value, showing infinitely many bound states.

## Contribution

It provides a rigorous mathematical proof of the Efimov effect for a specific three-particle system with mass-dependent conditions and zero-energy resonance.

## Key findings

- For mass ratio m < m*, infinitely many negative eigenvalues exist.
- For mass ratio m > m*, only finitely many negative eigenvalues.
- Asymptotic behavior of eigenvalues follows N(z) ~ C(m)|log|z|| as z approaches zero.

## Abstract

We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass $m$. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for $m<m^* = (13.607)^{-1}$, we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian $H$. More precisely, we prove that for $m>m^*$ the number of negative eigenvalues of $H$ is finite and for $m<m^*$ the number $N(z)$ of negative eigenvalues of $H$ below $z<0$ has the asymptotic behavior $N(z) \sim \mathcal C(m) |\log|z||$ for $z \rightarrow 0^-$. Moreover, we give an upper and a lower bound for the positive constant $\mathcal C(m)$.