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    Painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons

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    Author
    Forrester, PJ; Frankel, NE; Garoni, TM; Witte, NS
    Date
    2003-01-01
    Source Title
    COMMUNICATIONS IN MATHEMATICAL PHYSICS
    Publisher
    SPRINGER
    University of Melbourne Author/s
    Forrester, Peter; Frankel, Norman; Garoni, Timothy; WITTE, NICHOLAS
    Affiliation
    Mathematics And Statistics
    Metadata
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    Document Type
    Journal Article
    Citations
    Forrester, P. J., Frankel, N. E., Garoni, T. M. & Witte, N. S. (2003). Painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 238 (1-2), pp.257-285. https://doi.org/10.1007/s00220-003-0851-3.
    Access Status
    This item is currently not available from this repository
    URI
    http://hdl.handle.net/11343/26326
    DOI
    10.1007/s00220-003-0851-3
    Description

    C1 - Journal Articles Refereed

    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 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.
    Keywords
    Differential; Difference and Integral Equations; Condensed Matter Physics - Other ; Mathematical Sciences; Physical Sciences

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