We investigate the possibility that the BEC-like phenomena recently detected on two-dimensional ¯nite trapped systems consist of fragmented condensates. We derive and diagonalize the one-body density matrix of a two-dimensional isotropically trapped Bose gas at ¯nite temperature. For the ideal gas, the procedure reproduces the exact harmonic-oscillator eigenfunctions and the Bose distribution. We use a new collocation-minimization method to study the interacting gas in the Hartree-Fock approximation and obtain a ground-state wavefunction and condensate fraction consistent with those obtained by other methods. The populations of the next few eigenstates increase at the expense of the ground state but continue to be negligible; this supports the conclusion that two-dimensional BEC is into a single state. PACS numbers: 03.75.Hh, 05.30.Jp, 05.70.Fh, 32.80.Pj Bose-Einstein condensation (BEC) cannot occur at ¯nite temperature in a strictly two-dimensional trapped interacting gas in the thermodynamic limit.1 On the other hand, phenomena resembling BEC have been detecte
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