Trimers in the resonant 2+1 fermionic problem on a narrow Feshbach resonance : Crossover from Efimovian to Hydrogenoid spectrum

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 $alpha=m/M$. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within {sl each} sector of fixed total angular momentum $l$. For $alpha$ increasing from 0, we find that the trimer states first appear at the $l$-dependent Efimovian threshold $alpha_c^{(l)}$, where the Efimov exponent $s$ vanishes, and that the {sl entire} trimer spectrum (starting from the ground trimer state) is geometric for $alpha$ tending to $alpha_c^{(l)}$ from above, with a global energy scale that has a finite and non-zero limit. For further increasing values of $alpha$, the least bound trimer states still form a geometric spectrum, with an energy ratio $exp(2pi/|s|)$ 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.