Motivated by the recent article of Shea et al (2009 Am. J. Phys. 77 511) we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting via a short-range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two-dimensional problem is less trivial, requiring a more careful treatment as compared to the one-dimensional case. Our two-dimensional analysis reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two-dimensional pseudopotential.
We study bilayer fermionic cold-atom systems with dipolar interactions, as well as a two-component tunable pseudopotential (TCTP) model which keeps only the zeroth- and first-order Haldane pseudopotentials, at a total Landau-level filling factor of 1/2. Our numerical results on the TCTP model indicate that a Haldane-Rezayi state describes the critical point between strong and weak
d-wave pairing quantum Hall phases. Upon further increasing the attractive zeroth-order pseudopotentials, the system transits from the strong-pairing phase to a stripe phase and then to a cluster phase (or phase separation). The dipolar interaction can be mapped onto the TCTP model in the strong-pairing phase if high-order pseudopotentials are ignored. Our numerical results show that this is indeed the case, so the strong-pairing phase exists in the cold-atom system.