A Zero-Range Model for the Efimov Effect in the Born-Oppenheimer Approximation
G. Basti, D. Ferretti, A. Teta
TL;DR
The paper analyzes a three-body quantum system with zero-range interactions in three dimensions under the Born-Oppenheimer approximation. The fast dynamics at fixed separations yield a y-dependent bound $\mathcal{E}(|y|) = - \frac{( W(e^{\theta(|y|)}) - \theta(|y|) )^{2}}{\nu |y|^{2}}$, producing an effective slow Hamiltonian $h_{\mathcal{E}} = - \frac{1}{\mu} \Delta + \mathcal{E}(|y|)$. Solving the resulting s-wave radial problem reveals an infinite sequence of negative BO energies $E_{BO;n}$ accumulating at 0, with geometric spacing $E_{BO;n}/E_{BO;n+1} \to e^{2\pi/\beta}$ where $\beta = \sqrt{ (\mu/\nu) W(1)^2 - 1/4 }$. This establishes an Efimov-like spectrum in a rigorous zero-range, BO setting and extends prior results.
Abstract
In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].