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

##### 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

Show full item record##### 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

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##### 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 SciencesExport Reference in RIS Format

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