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Universal Two-Body Spectra of Ultracold Harmonically Trapped Atoms in Two and Three Dimensions

We consider the spectrum of two ultracold harmonically trapped atoms interacting via short-range interactions. The Green's function approach is used to unify the two and three dimensional cases. We derive criteria for the universality of the spectrum, i.e. its independence of the details of the short-range interaction. The results in three dimensions are examplified for narrow s-wave Feshbach resonances and we show how effective range corrections can modify the rearrangement of the level structure. However, this requires extremely narrow resonances or very tight traps that are not currently experimentally available. In the two-dimensional case we discuss the p-wave channel in detail and demonstrate how the non-universality of the spectrum arises within the Green's function approach. We then show that the spectrum is not particularly sensitive to the short-distance details in the case when the two-body interaction has a bound state.