Rigorous derivation of the Efimov effect in a simple model

Abstract

AbstractWe consider a system of three identical bosons in $$\mathbb {R}^3$$ R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $$ a > 0$$ 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 $$E_n$$ E n accumulating at zero and fulfilling the asymptotic geometrical law $$\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty $$ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where $$s_0\approx 1.00624$$ s 0 ≈ 1.00624 .