TY - JOUR AU - Forrester, PJ AU - Frankel, NE AU - Garoni, TM AU - Witte, NS Y2 - 2014/05/21 Y1 - 2003/01/01 SN - 0010-3616 UR - http://hdl.handle.net/11343/26326 AB - The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlev\'e VI transcendent in $\sigma$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlev\'e V and VI. We discuss how our results can be used to compute the ground state occupations. N1 - application/pdf LA - English PB - SPRINGER KW - Differential KW - Difference and Integral Equations; Condensed Matter Physics - Other ; Mathematical Sciences; Physical Sciences T1 - Painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons DO - 10.1007/s00220-003-0851-3 IS - COMMUNICATIONS IN MATHEMATICAL PHYSICS VL - 238 IS - 1-2 SP - 257-285 L1 - /bitstream/handle/11343/26326/116101_5162.pdf?sequence=1&isAllowed=n ER -