Efimov Three-Body States

Universal log-periodic tower: En+1/En = e−2π/s₀ ≈ 1/515 (equal mass), an+1/an ≈ 22.7

Efimov spectrum: bound state energies vs scattering length a
Hyperspherical wave function |Ψ(R)|² for selected state
515×
Energy ratio E_n/E_{n+1}
1.00624
s₀ (equal mass)
22.7
a_{n+1}/a_n
Efimov (1970): At unitarity (a→±∞), three identical bosons have an infinite tower of bound states with geometrically spaced energies: E_n = E_0 · e^{−2πn/s₀}, s₀ ≈ 1.00624 for equal masses.
Ratio: e^{2π/s₀} ≈ 515.03 in energy; e^{π/s₀} ≈ 22.7 in scattering length. First observed (2006) in ultracold Cs atoms near Feshbach resonance.
Discrete scale invariance: The Efimov spectrum is invariant under rescaling by λ = e^{π/s₀} ≈ 22.7 — the quantum three-body problem has a limit cycle (not a fixed point) under RG.