# Impenetrable bosons, random matrices, and generalized Fisher-Hartwig determinants.

#### by Tim Garoni

Institution: Institute for Mathematics and its Applications, University of Minnesota
Date: Thu 7th July 2005
Time: 2:15 PM
Location: Theatre 1, Old Geology Building, The University of Melbourne

Abstract: This talk will discuss a class of 1D many-body quantum
systems, the impenetrable'' bosons. There is an intimate connection
between these systems and random matrix theory, and many relevant
physical quantities of these systems can be computed exactly. There is
also an intimate connection with a class of determinants of a
generalized Fisher-Hartwig'' type. The asymptotics of these
determinants constitutes an interesting open problem in its own right.
We discuss the relevance of these determinants to the impenetrable
boson problem and to random matrix theory, and also discuss some
recent conjectures concerning the asymptotics of such determinants,
and proofs of special cases. Finally, using these asymptotic results
for the Fisher-Hartwig determinants, we show that for systems of
impenetrable bosons with certain boundary conditions the
eigenfunctions and eigenvalues of the limiting density matrix can be
computed exactly, thus giving a very complete characterization of
these systems.

For More Information: Iwan Jensen tel: 8344-5214 email: i.jensen@ms.unimelb.edu.au