The unitary three-body problem in a trap

Abstract

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.

Final version in English (4 pages) and in French (5 pages). Compared to the published version, corrects two errors (an error in the exponent on the right-hand side of equation (16) and the omission of an integer solution for fermions in footnote-in-bib [16]; see note 14 on page 95 and page 84 of Felix Werner's PhD thesis, version 3, available at https://theses.hal.science/tel-00285587v3)