A semiclassical analysis of the Efimov energy spectrum in the unitary limit

TL;DR

This paper uses semiclassical methods to analyze the Efimov energy spectrum at unitarity, revealing its geometric nature and confirming the accuracy of WKB approximations in describing Efimov states.

Contribution

It demonstrates that the Efimov spectrum can be derived from classical periodic orbits and confirms the effectiveness of WKB quantization in capturing its properties.

Findings

Efimov energy levels form a geometric spectrum generated by classical radial orbits.

WKB approximation accurately reproduces the Efimov spectrum and wavefunctions.

Efimov states' radii scale inversely with their energies.

Abstract

We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.