Rigorous derivation of the Efimov effect in a simple model

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, \emph{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 $E_{n+1}\mathrm{/}{E}_n\to e^{-\frac{2\pi }{s_0}}\text{for}n\to +\mathrm{\infty }$ holds, where $s_0\approx 1.00624$.