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 $\mathcal C(m)$.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.