Efimov Effect for a Three-Particle System with Two Identical Fermions

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 ) ∼ C ( m ) | log | z | | for z → 0 - . Moreover, we give an upper and a lower bound for the positive constant C ( m ) .