## A low-dimensional interacting quantum Bose gas amenable to exact treatment

Since the achievement of creating a Bose-Einstein condensate in the laboratory the possibility of
fabricating clean, low-dimensional quantum many-body systems with controlled, tunable interactions has come about.
With such control a tremendous advance in our understanding of strongly interacting and correlated quantum systems is now possible.
Therefore contact between experimental investigations and low-dimensional quantum systems amenable to rigorous theoretical
treatment has become a reality. One such system is the system of {$ N $} Bosons in one-dimension interacting with an infinitely
strong contact potential known as the Impenetrable Bose gas (IBG). It is one of the few systems where key quantities such as the density matrix
can be evaluated exactly for the ground state of the system. The density matrix is significant because,
the Onsager-Penrose criteria for Bose-Einstein condensation requires the lowest eigenvalue of this matrix, {$ \lambda_0(N) $}.

The IBG is related to a number of other systems, for example it is the infinite coupling
limit **Gir_1960** of the Lieb-Liniger Bose gas **LL_1963** which is soluble via the Bethe-ansatz
and is the quantised form of the classical Girardeau-Tonks gas **Mat_1993a**.
There is a collection of some of the early papers mentioned above in Chapter 5 of Mattis' book **Mat_1993a**
plus a nice summary of them in context at the beginning of this Chapter.
A more thorough theoretical treatment of the low-dimensional Bose gas is given in **LSSY_2005** and a more
focused review can be found in **SLY_2005**.

An important study of the density matrix was undertaken by Lenard **Len_1964**, and amongst a number of
results he found is the fact that the density matrix is the Fredholm minor of a particular integral
operator and using this he established rigorous results for {$ \lambda_0(N) $}.
He also extended the earlier Lieb-Liniger works to finite temperature in **Len_1966**.
The density matrix for the homogeneous IBG on a ring of circumference {$ L $} in the thermodynamic limit
{$ N, L \to \infty $} was found, in a famous 1980 result of Jimbo, Miwa, Mori and Sato **JMMS_1980**,
to be characterised by the solution of a particular case of Painlevé's fifth equation. As a consequence
one needs to solve a non-linear second-order differential equation in order to analyse the density matrix.

This result was generalised in 2003 by the author and collaborators **FFGW_2003** to the case of a finite
number of particles {$ N $} on a ring of size {$ L $}, although still a homogeneous system. In this case the sixth
Painlevé transcendent was crucial to understanding the system. In addition to a non-linear second-order
differential equation the density matrix can also be characterised by a non-linear recurrence relation in
the integer variable {$ N $}.

It is proposed to use a range of tools from integrable systems and orthogonal polynomial theory to analytically
study the density matrix of the IBG in these more realistic settings. This would suit a student with a
background in classical analysis who also has an interest in strongly interacting quantum many-body systems.

**FFGW_2003**
P.J. Forrester, N.E. Frankel, T.M. Garoni, and N.S. Witte. *Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable bosons*. Comm. Math. Phys., 238(1-2):257--285, 2003.

**Gir_1960**
M.D. Girardeau. *Relationship between systems of impenetrable bosons and fermions in one dimension*. J. Math. Phys., 1:516--523, 1960.

**JMMS_1980**
M. Jimbo, T. Miwa, Y. Môri, and M. Sato. *Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent*. Phys. D, 1(1):80--158, 1980.

**Len_1964**
A. Lenard. *Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons*. J. Math. Phys., 5(7):930--943, 1964.

**Len_1966**
A. Lenard. *One-dimensional impenetrable bosons in thermal equilibrium*. J. Math. Phys., 7(7):1268--1272, 1966.

**LL_1963**
E.H. Lieb and W. Liniger. *Exact analysis of an interacting Bose gas. I. The general solution and the ground state*. Phys. Rev. (2), 130:1605--1616, 1963.

**LSSY_2005**
E.H. Lieb, R. Seiringer, J.P. Solovej, and J. Yngvason. *The mathematics of the Bose gas and its condensation*, Vol.34 of Oberwolfach Seminars. Birkhäauser Verlag, Basel, 2005.

**Mat_1993a**
D.C. Mattis, editor. *The Many-Body Problem. An encyclopedia of exactly solved models in one dimension*. World Scientific Publishing Co., Inc., River Edge, NJ, 1993.

**SLY_2005**
R. Seiringer, E.H. Lieb, and J. Yngvason. *One-dimensional behaviour of dilute Bose gases in traps*. In XIVth International Congress on Mathematical Physics, pages 179--185. World Sci. Publ., Hackensack, NJ, 2005.