Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons |
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Authors: | PJ Forrester NE Frankel TM Garoni NS Witte |
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Institution: | (1) Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia. E-mail: P.Forrester@ms.unimelb.edu.au; N.Witte@ms.unimelb.edu.au, AU;(2) School of Physics, University of Melbourne, Victoria 3010, Australia. E-mail: n.frankel@physics.unimelb.edu.au; t.garoni@physics.unimelb.edu.au, AU |
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Abstract: | 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 Painlevé
VI transcendent in Σ-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 Painlevé V and VI. We discuss how our results can be used to compute the
ground state occupations.
Received: 24 July 2002 / Accepted: 26 January 2003
Published online: 13 May 2003
Communicated by L. Takhtajan |
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