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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 $α=m/M$. 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 $α$ increasing from 0, we find that the trimer states first appear at the $l$-dependent Efimovian threshold $α_c^{(l)}$, where the Efimov exponent $s$ vanishes, and that the {\sl entire} trimer spectrum (starting from the ground trimer state) is geometric for $α$ tending to $α_c^{(l)}$ from above, with a global energy scale that has a finite and non-zero limit. For further increasing values of $α$, 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.