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Applied Math Seminar The
finite one-dimensional Impenetrable Bose Gas and |
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The density matrix for the one-dimensional system of $
N $ impenetrable bosons at zero temperature on the circle has been found
as the $ \tau$-function of a particular member of the elliptic class of
Painlev\'e's sixth transcendent. The Painlev\'e transcendents are solutions
to a class of nonlinear differential equations first studied almost a
century ago which have the special property of Painlev\'e integrability
- i.e. the absence of movable branch points, and about which a lot is
currently known. A differential equation, of second-order and second degree,
is derived for the logarithmic derivative of the density matrix, the so-called
Jimbo-Miwa-Ueno $ \sigma-$function form. Using a result due to Lenard
(1964), that the density matrix also has a Toeplitz determinant form it
is possible to relate this problem to a system of orthogonal polynomials
on the unit circle with a generalised Jacobi weight. From this observation
a nonlinear third order difference equation for the density matrix in
the boson number $ N $ can be found. Utilising these results we can give
an exact expression for the growth of the momentum distribution $ n(k)
$ of the bosons near the central peak $ k=0 $ as $ N \to \infty $, i.e.
$ \rho_{N}(0) \propto \sqrt{N} $ which is of interest in the recent experiments
which confine a finite number of bosons in quasi-one dimensional traps.
Implications of our current knowledge of integrable systems with the Painlev\'e
property for one-dimensional systems of impenetrable bosons confined by
potentials, or in different geometries and subject to different boundary
conditions may be explored given time. |