Physics – Condensed Matter – Quantum Gases
Scientific paper
2009-11-16
Physics
Condensed Matter
Quantum Gases
No figures. Accepted for publication in J. Phys. A: Math. Theor
Scientific paper
Motivated by the recent article of P. Shea {\it et al.} [Am. J. Phys. {\bf 77} (6), 2009] we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting {\it via} a short range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, provided that the range of the potential is much smaller than the oscillator length, 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 likewise reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two dimensional pseudopotential.
Farrell Aaron
van Zyl Brandon P.
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