Trimers in the resonant (2 + 1)-fermion problem on a narrow Feshbach resonance: Crossover from Efimovian to hydrogenoid spectrum

摘要

We study the quantum three-body free-space problem of two same-spin-state fermions of mass m interacting with a different particle of mass M, on an infinitely narrow Feshbach resonance with infinite s-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio a = m/M. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within each sector of fixed total angular momentum 1. For a increasing from 0, we find that the trimer states first appear at the 1-dependent Efimovian threshold a_c~(l), where the Efimov exponent s vanishes, and that the entire trimer spectrum (starting from the ground trimer state) is geometric for a tending to a_c~(1) from above, with a global energy scale that has a finite and nonzero limit. For further increasing values of a, the least bound trimer states still form a geometric spectrum, with an energy ratio exp(2π/|s|) that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.