Rigorous derivation of the Efimov effect in a simple model

Abstract

We consider a system of three identical bosons in \(\mathbb {R}^3\) with two-body zero-range interactions and a three-body hard-core repulsion of a given radius \( a > 0\). Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues \(E_n\) accumulating at zero and fulfilling the asymptotic geometrical law \(\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty \) holds, where \(s_0\approx 1.00624\).